具有正初始能量p⁃Laplace方程解的爆破

具有正初始能量 p ⁃Laplace方程解的爆破 Blow-up of Solutions for p ⁃Laplace Equation with Positive Initial Energy 1 first-author 吴秀兰，女，副教授，现从事偏微分方程研究。E-mail：chjlsywxl@126.com 吴秀兰，女，副教授，现从事偏微分方程研究。E-mail：chjlsywxl@126.com chjlsywxl@126.com 2 3 corresp E-mail: jlfj8080@126.com E-mail: jlfj8080@126.com jlfj8080@126.com 长春理工大学 理学院， 吉林 长春 130012 长春理工大学 理学院， 吉林 长春 130012 School of Science, Changchun University of Science and Technology, Changchun 130012， Jilin, China School of Science, Changchun University of Science and Technology, Changchun 130012， Jilin, China 长春工业大学 机电工程学院， 吉林 长春 130012 长春工业大学 机电工程学院， 吉林 长春 130012 School of Mechatronic Engineering, Changchun University of Technology, Changchun 130012, Jilin, China School of Mechatronic Engineering, Changchun University of Technology, Changchun 130012, Jilin, China 吉林师范大学 数学学院， 吉林 四平 136000 吉林师范大学 数学学院， 吉林 四平 136000 Institute of Mathematics, Jinlin Normal University, Siping 136000， Jilin, China Institute of Mathematics, Jinlin Normal University, Siping 136000， Jilin, China 研究了具有齐次Dirichlet边界条件 p ⁃Laplace方程解的爆破问题。利用构造凸函数方法证明了初始能量为正数且 <math id="M1"><mi>r</mi><mo>></mo><mi>p</mi><mo stretchy="false">(</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419270&type= 17.35666656 3.30199981 条件下，解在有限时刻爆破。 This paper deals with homogeneous Dirichlet boundary value problem of a class of p -Laplace equations. A blow-up result is established when initial energy is positive and <math id="M2"><mi>r</mi><mo>></mo><mi>p</mi><mo stretchy="false">(</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419273&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419272&type= 17.35666656 3.30199981 making use of constructing concave function. 爆破 p ⁃Laplace方程 正初始能量 blow-up p -Laplace equation positive initial energy 0　 引 言 1 考虑如下 p -Laplace方程 <math display="block" id="M3"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">v</mi><mo stretchy="false">(</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mo>∇</mo><mi>u</mi></mrow></mfenced></mrow><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mo>∂</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419275&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419274&type= 71.71266174 16.84866714 其中， <math id="M4"><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419276&type= 23.11400032 3.89466667 ， <math id="M5"><mo>∂</mo><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419278&type= 4.14866638 2.79399991 是光滑， <math id="M6"><mi>f</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>C</mi><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419280&type= 19.64266586 3.81000018 且满足如下条件 <math display="block" id="M7"><mi>s</mi><mi>f</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>r</mi><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>≥</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>s</mi></mrow></mfenced></mrow><mrow><mi>r</mi></mrow></msup><mo>,</mo><mi>r</mi><mo>></mo><mi>p</mi><mo>></mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419282&type= 48.09066772 4.48733330 其中， <math id="M8"><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><mi mathvariant="normal">d</mi><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419285&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419284&type= 27.26266861 6.77333355 。 p -Laplace方程源于非线性动力学，关于该问题解存在性、唯一性结果非常完善，见文献[ 1 1 , 2 2 ]。 现实生活中很多实际问题用 p -Laplace方程刻画更符合实际。关于 p -Laplace方程解爆破问题研究，近年来引起国内外广泛关注，见文献[ 3 3 , 4 4 , 5 5 ]。1993年，Zhao等 [ 6 6 ] 证明了初始能量为负数时，解在有限时刻爆破。2008年Liu等 [ 7 7 ] 证明了初始能量大于某个正数且 <math id="M9"><mi>r</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mfrac><mrow><mi>n</mi><mi>p</mi></mrow><mrow><mi>n</mi><mo>-</mo><mi>p</mi></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo><</mo><mi>p</mi><mo><</mo><mi>n</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419287&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419286&type= 40.55533218 7.95866680 时解在有限时刻爆破。本文借助文献[ 8 8 ]思想，通过构造凸函数方法，推广文献[ 7 7 ]的结果（ <math id="M10"><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419289&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419288&type= 8.55133343 3.30199981 ）。 引进能量泛函 <math display="block" id="M11"><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><msup><mrow><msub><mo>∫</mo><mrow><mi mathvariant="normal">Ω</mi></mrow></msub><mrow><mfenced open="|" close="|" separators="|"><mrow><mo>∇</mo><mi>u</mi></mrow></mfenced></mrow></mrow><mrow><mi>p</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>-</mo><msub><mo>∫</mo><mrow><mi mathvariant="normal">Ω</mi></mrow></msub><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419291&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419290&type= 53.67866516 8.29733276 则 <math display="block" id="M12"><mi>E</mi><mi>'</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">d</mi><mi>x</mi><mo>≤</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419292&type= 30.48000145 4.82599974 引理1 [ 9 9 ] 假设 <math id="M13"><mi>θ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419295&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419294&type= 6.68866634 3.55599999 是二次连续可微的函数且有如下不等式成立 <math display="block" id="M14"><mi>θ</mi><mo>″</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>θ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>θ</mi><mi>'</mi><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419297&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419296&type= 50.03800201 4.23333359 ， <math display="block" id="M15"><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419298&type= 7.95866680 2.79399991 其中 <math id="M16"><mi>β</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419301&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419300&type= 8.72066689 3.81000018 为常数。如果 <math id="M17"><mi>θ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419302&type= 15.07066631 3.55599999 <math id="M18"><mi>θ</mi><mi>'</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419304&type= 14.64733315 3.55599999 则 <math id="M19"><mo>∃</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><mn mathvariant="normal">0</mn><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mfrac><mrow><mi>θ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow><mi>β</mi><mi>θ</mi><mi>'</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419307&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419306&type= 34.37466812 8.63599968 ，使得 <math id="M20"><mi>θ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">∞</mi><mo>,</mo><mi>t</mi><mo>→</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419308&type= 26.75466537 4.48733330 。 1　 主要结论及证明 1 定理1 若 <math id="M21"><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419310&type= 10.83733368 3.55599999 是问题(1)的解且对于 <math id="M22"><mo>∀</mo><mi>r</mi><mo>></mo><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419312&type= 10.49866676 3.72533321 ， <math id="M23"><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419315&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419314&type= 16.42533302 3.55599999 初值 <math id="M24"><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419317&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419316&type= 3.04799986 3.55599999 满足 <math display="block" id="M25"><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>></mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>r</mi><mi>p</mi><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow><mi>r</mi><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></msup><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>Ω</mi></mrow></mfenced></mrow><mrow><mfrac><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419319&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419318&type= 41.06333542 10.92199993 则解 <math id="M26"><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419321&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419320&type= 10.83733368 3.55599999 在有限时刻爆破。 证 (4)式两端在 <math id="M27"><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419322&type= 8.04333305 3.55599999 上积分有 <math display="block" id="M28"><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub><mfenced separators="|"><mrow><mi>τ</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mi mathvariant="normal">d</mi><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419324&type= 49.02199936 6.77333355 问题(1)第一式两端同乘 <math id="M29"><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419326&type= 1.86266661 2.62466669 ，并在 <math id="M30"><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419328&type= 2.45533323 2.62466669 上积分有 <math display="block" id="M31"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mi>u</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419330&type= 40.72466660 7.45066643 <math display="block" id="M32"><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mi>u</mi><mfenced separators="|"><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">v</mi><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mo>∇</mo><mi>u</mi></mrow></mfenced></mrow><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419332&type= 51.98533249 6.68866634 <math display="block" id="M33"><mo>-</mo><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mo>∇</mo><mi>u</mi></mrow></mfenced></mrow><mrow><mi>p</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>+</mo><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mi>u</mi><mi>f</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419334&type= 43.01066971 6.68866634 下面分两种情况讨论： 情形1 若 <math id="M34"><mo>∀</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419336&type= 10.15999985 2.96333337 ， <math id="M35"><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419338&type= 14.05466652 3.55599999 ，由(5)及(6)式有 <math display="block" id="M36"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><mo>-</mo><mi>p</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mi>u</mi><mi>f</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">d</mi><mi>x</mi><mo>-</mo><mi>p</mi><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mi mathvariant="normal">d</mi><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419341&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419340&type= 82.46533203 7.45066643 结合(2)及(7)式进一步有 <math display="block" id="M37"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>≥</mo><mo>-</mo><mi>p</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>r</mi></mrow><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mi mathvariant="normal">d</mi><mi>x</mi><mo>≥</mo><mo>-</mo><mi>p</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mi>r</mi><mo>-</mo><mi>p</mi></mrow><mrow><mi>r</mi></mrow></mfrac><msup><mrow><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow></mrow><mrow><mi>r</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419342&type= 74.25267029 16.76399994 由Young不等式，有 <math display="block" id="M38"><msup><mrow><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mi>r</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>≥</mo><mi>λ</mi><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>-</mo><msup><mrow><mi>λ</mi></mrow><mrow><mfrac><mrow><mi>r</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mfrac><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></mfrac><mfenced open="|" close="|" separators="|"><mrow><mi>Ω</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419344&type= 69.17266846 10.24466705 其中， <math id="M39"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419347&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419346&type= 1.77800000 2.70933342 为待定正数。 综合(8)及(9)式有 <math display="block" id="M40"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>≥</mo><mo>-</mo><mi>p</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mo>+</mo><mfrac><mrow><mi>r</mi><mo>-</mo><mi>p</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mfenced separators="|"><mrow><mi>λ</mi><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>-</mo><msup><mrow><mi>λ</mi></mrow><mrow><mfrac><mrow><mi>r</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mfrac><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></mfrac><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mfenced open="|" close="|" separators="|"><mrow><mi>Ω</mi></mrow></mfenced></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419349&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419348&type= 105.41000366 12.86933327 取 <math id="M41"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419351&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419350&type= 1.94733334 2.62466669 满足如下条件 <math display="block" id="M42"><mn mathvariant="normal">1</mn><mo><</mo><mi>α</mi><mo><</mo><mfrac><mrow><mi>r</mi><mo>-</mo><mi>p</mi></mrow><mrow><mi>r</mi><mi>p</mi><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow></mfrac><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>Ω</mi></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mo>-</mo><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419352&type= 45.21199799 8.21266651 结合(9),(10)及(5)式,有 <math display="block" id="M43"><mfrac><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mo>≥</mo><mo>-</mo><mn mathvariant="normal">2</mn><mi>p</mi><mi>α</mi><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">2</mn><mi>p</mi><mi>α</mi><msubsup><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">d</mi><mi>τ</mi><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419354&type= 73.82933807 7.95866680 <math display="block" id="M44"><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>λ</mi></mrow><mrow><mi>r</mi></mrow></mfrac><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419356&type= 28.44799995 7.36599970 <math display="block" id="M45"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419359&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419358&type= 3.72533321 2.62466669 <math display="block" id="M46"><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>r</mi></mrow></mfrac><msup><mrow><mi>λ</mi></mrow><mrow><mfrac><mrow><mi>r</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mfrac><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></mfrac><mfenced open="|" close="|" separators="|"><mrow><mi>Ω</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419361&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419360&type= 49.19132996 7.36599970 由(12)式解得 <math display="block" id="M47"><msubsup><mrow><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>≥</mo><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>λ</mi><mi>t</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></msup><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419363&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419362&type= 49.69933319 7.87400007 <math display="block" id="M48"><mfrac><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>λ</mi></mrow></mfrac><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>λ</mi><mi>t</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></msup></mrow></mfenced><mo>×</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419365&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419364&type= 42.16399765 8.63599968 <math display="block" id="M49"><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi>p</mi><mi>α</mi><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>λ</mi></mrow><mrow><mfrac><mrow><mi>r</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mfrac><mrow><mfenced separators="|"><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfenced><mfenced separators="|"><mrow><mi>r</mi><mo>-</mo><mi>p</mi></mrow></mfenced></mrow><mrow><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfenced open="|" close="|" separators="|"><mrow><mi>Ω</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419367&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419366&type= 69.76533508 12.86933327 定义函数 <math id="M50"><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi></mrow></mfenced></mrow></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">d</mi><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419369&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419368&type= 25.82333374 7.95866680 ，假设函数 <math id="M51"><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419370&type= 10.83733368 3.55599999 整体存在，则 <math id="M52"><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419373&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419372&type= 6.77333355 3.81000018 是有界函数且有 <math display="block" id="M53"><mi>y</mi><mi>'</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>,</mo><mi>y</mi><mo>″</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419375&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419374&type= 45.97399902 7.45066643 将（13）式代入(12)式，有 <math display="block" id="M54"><mi>y</mi><mo>″</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">2</mn><mi>p</mi><mi>α</mi><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mi mathvariant="normal">d</mi><mi>τ</mi><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>λ</mi><mi>t</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></msup><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>λ</mi></mrow><mrow><mi>r</mi></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mn mathvariant="normal">2</mn><mi>p</mi><mi>α</mi><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfenced open="|" close="|" separators="|"><mrow><mi>Ω</mi></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>λ</mi></mrow><mrow><mfrac><mrow><mi>r</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419377&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419376&type= 120.56533051 21.50533485 考虑到(11)式,取 <math id="M55"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419378&type= 1.43933344 2.62466669 满足下列条件 <math display="block" id="M56"><mn mathvariant="normal">0</mn><mo><</mo><mi>ε</mi><mo><</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419381&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419380&type= 12.27666664 2.79399991 <math display="block" id="M57"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi><mi>α</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>r</mi></mrow></mfrac><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>Ω</mi></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mo>-</mo><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>-</mo><mn mathvariant="normal">2</mn><mi>p</mi><mi>α</mi><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419382&type= 68.83399963 10.92199993 定义函数 <math id="M58"><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>ε</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>c</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419385&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419384&type= 50.96933365 5.84200001 ，其中 <math id="M59"><mi>c</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419386&type= 1.52399993 2.62466669 满足 <math display="block" id="M60"><mi>c</mi><mo>></mo><mfrac><mrow><msup><mrow><mi>ε</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419388&type= 21.08199883 7.45066643 因此有 <math display="block" id="M61"><mi>φ</mi><mi>'</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mi>y</mi><mo>+</mo><msup><mrow><mi>ε</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mi>y</mi><mi>'</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419390&type= 45.46599960 5.84200001 <math display="block" id="M62"><mi>φ</mi><mo>″</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mi>y</mi><mo>+</mo><msup><mrow><mi>ε</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mi>y</mi><mo>″</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>y</mi><mi>'</mi><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419393&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419392&type= 65.70133209 5.84200001 令 <math id="M63"><mi>δ</mi><mo>=</mo><mn mathvariant="normal">4</mn><mi>c</mi><mo>-</mo><msup><mrow><mi>ε</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419394&type= 28.36333275 5.84200001 。由（15）式， <math id="M64"><mi>δ</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419396&type= 8.63599968 2.79399991 。由(16)式,有 <math display="block" id="M65"><mo stretchy="false">[</mo><mi>φ</mi><mi>'</mi><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><mn mathvariant="normal">4</mn><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>δ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>y</mi><mi>'</mi><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419399&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419398&type= 48.59866714 4.06400013 由恒等式 <math id="M66"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mi>u</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mi mathvariant="normal">d</mi><mi>x</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419401&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419400&type= 36.83000183 7.45066643 ，将其在 <math id="M67"><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419402&type= 8.04333305 3.55599999 上积分，有 <math display="block" id="M68"><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">2</mn><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">d</mi><mi>x</mi><mi mathvariant="normal">d</mi><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419405&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419404&type= 64.09266663 6.77333355 即有 <math display="block" id="M69"><mo stretchy="false">[</mo><mi>y</mi><mi>'</mi><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msup><mrow><mo>+</mo><mn mathvariant="normal">2</mn><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></mrow><mi mathvariant="normal">d</mi><mi>x</mi><mi mathvariant="normal">d</mi><mi>τ</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419406&type= 91.10133362 6.77333355 <math display="block" id="M70"><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">4</mn><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">d</mi><mi>τ</mi></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo>⋅</mo><msup><mrow><mfenced separators="|"><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">d</mi><mi>τ</mi></mrow></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo>+</mo><mn mathvariant="normal">4</mn><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub><mfenced separators="|"><mrow><mi>τ</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">d</mi><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419409&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419408&type= 63.83866501 22.69066620 由带 <math id="M71"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419411&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419410&type= 1.43933344 2.62466669 的Cauchy不等式，上式化为 <math display="block" id="M72"><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>y</mi><mi>'</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≤</mo><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">4</mn><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msubsup><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">d</mi><mi>τ</mi><mo>+</mo><mn mathvariant="normal">2</mn><mi>ε</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>ε</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msubsup><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">d</mi><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419412&type= 68.07199860 17.77999878 结合(14),(17)和(18)式，有 <math display="block" id="M73"><mn mathvariant="normal">2</mn><mi>φ</mi><mo>″</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mi>p</mi><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>φ</mi><mi>'</mi><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≥</mo><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>ε</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">2</mn><mi>p</mi><mi>α</mi><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">d</mi><mi>τ</mi></mrow><mo>⋅</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>λ</mi><mi>t</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></msup><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>λ</mi></mrow><mrow><mi>r</mi></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mn mathvariant="normal">2</mn><mi>p</mi><mi>α</mi><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfenced open="|" close="|" separators="|"><mrow><mi>Ω</mi></mrow></mfenced><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>λ</mi></mrow><mrow><mfrac><mrow><mi>r</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow></mfenced></mrow></mfenced><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419415&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419414&type= 174.41333008 18.54199982 <math display="block" id="M74"><mn mathvariant="normal">2</mn><mi>p</mi><mi>α</mi><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mfenced open="{" close="}" separators="|"><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">4</mn><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">d</mi><mi>τ</mi><mo>+</mo><mn mathvariant="normal">2</mn><mi>ε</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>ε</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">d</mi><mi>τ</mi></mrow></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419416&type= 125.72999573 7.87400007 定义函数 <math display="block" id="M75"><mi>H</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>λ</mi></mrow><mrow><mi>r</mi></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mn mathvariant="normal">2</mn><mi>p</mi><mi>α</mi><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfenced open="|" close="|" separators="|"><mrow><mi>Ω</mi></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>λ</mi></mrow><mrow><mfrac><mrow><mi>r</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419419&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419418&type= 61.72200012 19.47333336 因此 <math display="block" id="M76"><mi>H</mi><mi>'</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>r</mi></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>r</mi></mrow></mfrac><mfenced open="|" close="|" separators="|"><mrow><mi>Ω</mi></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>λ</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419421&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419420&type= 45.38133240 19.47333336 令 <math id="M77"><mi>H</mi><mi>'</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419422&type= 15.91733360 3.55599999 ，有 <math id="M78"><msub><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msubsup><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>Ω</mi></mrow></mfenced></mrow><mrow><mo>-</mo><mfrac><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419425&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419424&type= 38.69266510 7.11199999 。经计算 <math display="block" id="M79"><mtable columnalign="left"><mtr><mtd><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi>λ</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></munder><mi>H</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>r</mi></mrow></mfrac><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>Ω</mi></mrow></mfenced></mrow><mrow><mo>-</mo><mfrac><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><mi>p</mi><mi>α</mi><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419426&type= 68.58000183 14.05466652 取 <math id="M80"><mi>λ</mi><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419429&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419428&type= 11.68400002 3.64066648 ，考虑 <math id="M81"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419430&type= 1.43933344 2.62466669 取值及 <math id="M82"><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>λ</mi><mi>t</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></msup><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419432&type= 18.11866570 5.24933338 ， <math id="M83"><mi>φ</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419435&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419434&type= 8.88999939 3.64066648 ，有 <math display="block" id="M84"><mn mathvariant="normal">2</mn><mi>φ</mi><mo>″</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mi>p</mi><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced><mo stretchy="false">[</mo><mi>φ</mi><mi>'</mi><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≥</mo><mn mathvariant="normal">2</mn><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>ε</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi>p</mi><mi>α</mi><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub><mfenced separators="|"><mrow><mi>τ</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">d</mi><mi>τ</mi><mo>+</mo><mi>p</mi><mi>ε</mi><mi>α</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419436&type= 144.01800537 8.29733276 <math display="block" id="M85"><mo>-</mo><mn mathvariant="normal">2</mn><mi>p</mi><mi>α</mi><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419439&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419438&type= 16.42533302 3.72533321 <math display="block" id="M86"><mfenced open="{" close="" separators="|"><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">4</mn><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">d</mi><mi>τ</mi></mrow></mrow></mfenced><mo>+</mo><mn mathvariant="normal">2</mn><mi>ε</mi><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419441&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419440&type= 67.98733521 7.87400007 <math display="block" id="M87"><mfenced open="" close="}" separators="|"><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi>ε</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">d</mi><mi>τ</mi></mrow></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419443&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419442&type= 45.29666519 7.87400007 因此有 <math display="block" id="M88"><mi>φ</mi><mo>″</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mi>p</mi><mi>α</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac></mrow></mfenced><mo stretchy="false">[</mo><mi>φ</mi><mi>'</mi><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419445&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419444&type= 62.82266617 8.38199997 由于 <math id="M89"><mi>φ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419447&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419446&type= 15.91733360 3.72533321 <math id="M90"><mi>φ</mi><mi>'</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419448&type= 14.98599911 3.72533321 ，应用引理1，我们有 <math id="M91"><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419450&type= 19.30399895 3.72533321 (当 <math id="M92"><mi>t</mi><mo>→</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>φ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>p</mi><mi>α</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mi>φ</mi><mi>'</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419453&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419452&type= 39.11599731 8.46666718 )。由于 <math id="M93"><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419454&type= 7.02733326 3.72533321 关于 <math id="M94"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419457&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419456&type= 1.86266661 3.55599999 是连续的且 <math id="M95"><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419459&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419458&type= 6.77333355 3.81000018 整体存在，故与 <math id="M96"><mi>φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">(</mo><mi>t</mi><mo>→</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419461&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419460&type= 32.17333221 4.65666676 矛盾。 情形2 若 <math id="M97"><mo>∃</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419463&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419462&type= 13.03866673 4.23333359 使得 <math id="M98"><mi>E</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≤</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419465&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419464&type= 20.31999969 4.57200003 。令 <math id="M99"><mi>v</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419467&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419466&type= 32.93533325 4.48733330 ，则有 <math id="M100"><mi>E</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419469&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419468&type= 31.91933250 4.48733330 。由(5)式，我们有 <math display="block" id="M101"><mi>E</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≤</mo><mi>E</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≤</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419471&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419470&type= 38.09999847 3.64066648 定义函数 <math id="M102"><mi>G</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419473&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419472&type= 33.35866928 6.68866634 ，那么 <math display="block" id="M103"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mi>v</mi><mfenced separators="|"><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">v</mi><mfenced separators="|"><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mo>∇</mo><mi>v</mi></mrow></mfenced></mrow><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mo>∇</mo><mi>v</mi></mrow></mfenced><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mfenced><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419475&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419474&type= 88.39199829 7.45066643 <math display="block" id="M104"><mo>-</mo><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mo>∇</mo><mi>v</mi></mrow></mfenced></mrow><mrow><mi>p</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>+</mo><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mi>v</mi><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mrow><mo>≥</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419476&type= 47.75199890 6.68866634 <math display="block" id="M105"><mo>-</mo><mi>p</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mi>r</mi><mo>-</mo><mi>p</mi></mrow><mrow><mi>r</mi></mrow></mfrac><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>v</mi></mrow></mfenced></mrow><mrow><mi>r</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419478&type= 41.06333542 7.11199999 结合(21)及（22）式，有 <math display="block" id="M106"><mi>G</mi><mi>'</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>r</mi></mrow></mfrac><msup><mrow><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>v</mi></mrow></mfenced></mrow></mrow><mrow><mi>r</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419482&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419480&type= 42.33333206 7.45066643 由嵌入定理 <math id="M107"><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup><mo stretchy="false">(</mo><mi>Ω</mi><mo stretchy="false">)</mo><mover accent="false"><mo stretchy="true">→</mo><mrow><mi mathvariant="normal">嵌</mi><mi mathvariant="normal">入</mi></mrow></mover><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>Ω</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419483&type= 35.72933197 4.40266657 有 <math display="block" id="M108"><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>v</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><mi>B</mi><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>v</mi></mrow></mfenced></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419486&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419485&type= 22.01333427 4.65666676 这里 <math id="M109"><mi>B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419487&type= 2.53999996 2.62466669 是嵌入常数，结合(23)及（24）式，有 <math display="block" id="M110"><mi>G</mi><mi>'</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>r</mi></mrow></mfrac><msup><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msup><msubsup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi>v</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>r</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>r</mi></mrow></mfrac><mo stretchy="false">[</mo><mi>G</mi><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mfrac><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419490&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419489&type= 78.57066345 7.36599970 利用Gronwall不等式，有 <math display="block" id="M111"><msup><mrow><mi>G</mi></mrow><mrow><mfrac><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>G</mi></mrow><mrow><mfrac><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mi>r</mi></mrow></mfrac><msup><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msup><mi>t</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419492&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419491&type= 66.71733856 11.93799973 从上面不等式得 <math display="block" id="M112"><mi>G</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">∞</mi><mo>,</mo><mi>t</mi><mo>→</mo><msup><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>≤</mo><mfrac><mrow><mi>r</mi><msup><mrow><mi>G</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mo>-</mo><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>r</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419494&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419493&type= 61.89133453 10.49866676 综合以上两种情况，有 <math id="M113"><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419496&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419495&type= 10.83733368 3.55599999 在有限时刻爆破。 参考文献 DIBENEDETTO E . Degenerate Parabolic Equations [M]. Berlin : Springer , 1993 ： 41 - 75 . Degenerate Parabolic Equations WU Z Q , ZHAO J N , YIN J X , et al . Nonlinear Diffusion Equations [M]. Singapore : World Scientific , 2001 : 147 - 266 . Nonlinear Diffusion Equations LIAO M L , GAO W J . Blow-up phenomena for a nonlocal p -Laplace equation with Neumann boundary conditions [J]. Archiv der Mathematik , 2017 , 108 ( 3 ): 313 - 324 . DOI: 10.1007/s00013-016-0986-z http://dx.doi.org/10.1007/s00013-016-0986-z . Blow-up phenomena for a nonlocal p-Laplace equation with Neumann boundary conditions HAN Y Z . A class of fast diffusion p -Laplace equation with arbitrarily high initial energy [J]. Comptes Rendus Mecanique , 2018 , 346 ( 12 ): 1153 - 1158 . DOI: 10.1016/j.crme.2018.06.013 http://dx.doi.org/10.1016/j.crme.2018.06.013 . A class of fast diffusion p-Laplace equation with arbitrarily high initial energy WU X L . The blow-up of solutions for m -Laplacian equations with variable sources under positive initial energy [J]. Computers & Mathematics with Applications , 2016 , 72 ( 9 ): 2516 - 2524 . DOI: 10.1016/j.camwa.2016.09.015 http://dx.doi.org/10.1016/j.camwa.2016.09.015 . The blow-up of solutions for m-Laplacian equations with variable sources under positive initial energy ZHAO J N . Existence and nonexistence of solutions for <math id="M114"><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">v</mi><mfenced separators="|"><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mo>∇</mo><mi>u</mi></mrow></mfenced></mrow><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mo>∇</mo><mi>u</mi></mrow></mfenced><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mo>∇</mo><mi>u</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419497&type= 48.34466553 5.07999992 [J]. Journal of Mathematical Analysis and Applications , 1993 , 172 : 130 - 146 . DOI: 10.1006/jmaa.1993.1012 http://dx.doi.org/10.1006/jmaa.1993.1012 . Existence and nonexistence of solutions for ut=div∇up-2∇u+f(∇u,u,x,t) LIU W J , WANG M X . Blow-up of the solution for a p -Laplacian equation with positive initial energy [J]. Acta Applicandae Mathematicae , 2008 , 103 ( 2 ): 141 - 146 . DOI: 10.1007/s10440-008-9225-3 http://dx.doi.org/10.1007/s10440-008-9225-3 . Blow-up of the solution for a p-Laplacian equation with positive initial energy KHELGHATI A , KHADIJEH B . Blow-up phenomena for a nonlocal semilinear parabolic equation with positive initial energy [J]. Computers & Mathematics with Applications , 2015 , 70 ( 5 ): 896 - 902 . DOI: 10.1016/j.camwa.2015.06.003 http://dx.doi.org/10.1016/j.camwa.2015.06.003 . Blow-up phenomena for a nonlocal semilinear parabolic equation with positive initial energy LEVINE H A . Some nonexistence and instability theorems for solution of formally prabolic equations of the form <math id="M115"><mi>P</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>-</mo><mi>A</mi><mi>u</mi><mo>+</mo><mi>F</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419500&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419499&type= 25.56933403 3.64066648 [J]. Archive for Rational Mechanics and Analysis , 1973 , 51 ( 5 ): 371 - 386 . DOI: 10.1007/BF00263041 http://dx.doi.org/10.1007/BF00263041 . Some nonexistence and instability theorems for solution of formally prabolic equations of the form Put=-Au+F(u) 10.14188/j.1671-8836.2018.0141 O175.8 A 1671-8836（2020）03-0233-04 WU Xiulan,LI Wanting,FU Jun.Blow-up of Solutions for p -Laplace Equation with Positive Initial Energy [J]. J Wuhan Univ （ Nat Sci Ed ）,2020,66(3):233-236. DOI: 10.14188/j.1671-8836.2018.0141 http://dx.doi.org/10.14188/j.1671-8836.2018.0141 (Ch). WU Xiulan,LI Wanting,FU Jun.Blow-up of Solutions for p-Laplace Equation with Positive Initial Energy [J].J Wuhan Univ（Nat Sci Ed）,2020,66(3):233-236. DOI:10.14188/j.1671-8836.2018.0141(Ch). 吴秀兰,李婉婷,付军.具有正初始能量 p ⁃Laplace方程解的爆破[J].武汉大学学报（理学版）,2020,66(3):233-236. DOI:10.14188/j.1671-8836.2018.0141-吴秀兰 吴秀兰,李婉婷,付军.具有正初始能量p⁃Laplace方程解的爆破[J].武汉大学学报（理学版）,2020,66(3):233-236. DOI:10.14188/j.1671-8836.2018.0141-吴秀兰 国家青年自然科学基金（11301211）；吉林省教育厅十三五科学技术研究项目(2016211)；吉林师范大学博士启动基金（2014008） 2018-04-22 2020-06-24 2020-09-30 武汉大学学报（理学版） 3 66