一类具有非线性阻尼和多源项的波动方程

一类具有非线性阻尼和多源项的波动方程 A Class of Wave Equation with Nonlinear Damping and Multiple Terms first-author 晋守博, 男, 讲师, 主要从事偏微分方程的研究. E-mail:jin_shoubo@163.com 晋守博, 男, 讲师, 主要从事偏微分方程的研究. E-mail: jin_shoubo@163.com jin_shoubo@163.com 宿州学院 数学与统计学院, 安徽 宿州 234000 宿州学院 数学与统计学院, 安徽 宿州 234000 School of Mathematics and Statistics, Suzhou University, Suzhou 234000, Anhui, China School of Mathematics and Statistics, Suzhou University, Suzhou 234000, Anhui, China 给出了一类同时具有非线性阻尼和多个非线性源项的波动方程, 利用Gakerkin方法构造了方程的近似解, 并通过一些先验估计分析了近似解在不同空间的收敛性, 证明了该波动方程存在一个整体弱解.通过构造等价泛函的方法, 在特殊情况下, 研究方程整体解的渐近稳定性. The wave equation with nonlinear damping and nonlinear source terms is given in this paper. The approximate weak solutions of the problems are constructed by the Galerkin method, and the convergences are analyzed in some different spaces. Then, the existence of global weak solutions is obtained. Finally, the asymptotic behaviors of global solutions are investigated under some special assumptions. 非线性阻尼 波动方程 整体解 渐近性 nonlinear damping wave equation global solution asymptotic behavior 引言 1 本文研究一类同时具有非线性阻尼项和多个非线性源项的波动方程的初边值问题，具体形式如下: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4464988&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4464988&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4464988&type=middle 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465148&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465148&type=middle 为有界光滑区域，a为正常数，整数m > 2，非线性源项 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465151&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465151&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465151&type=middle u ，这里常数 a i ,b j ≥0,p i ,q j 满足： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4464994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4464994&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4464994&type=middle 具有阻尼项的非线性波动方程是一类非常重要的波动方程，它们是从具有黏性效应物体的运动中提出的，阻尼是指阻碍物体的相对运动并把运动能量转化为热能或其他可以耗散能量的一种作用.阻尼现象广泛存在于自然界中，例如弹性杆的纵向运动、具有表面张力的水波运动以及导体在变化的磁场中运动等都包含阻尼现象.近年来，越来越多的学者开始研究不同类型的波动方程解的存在惟一性问题，能量估计法是最重要的分析工具之一，研究表明对于含有正定能量的波动方 u tt －Δu=－|u| p－1 u ，利用Gakerkin近似解法可以有效解决，当方程含有非正定能量时，仅用该方法不能解决.1968年Sattinger通过位势井方法研究了含有非正定能量的波动方程整体解的存在性 [ 1 1 ] .1975年，Payne和Sattinger证明不具有正定能量的波动方程 u tt －Δu=|u| p－1 u 整体解的存在性与爆破现象 [ 2 2 ] ，2007年叶朝辉分析了具有两个异号源项的波动方程 u tt －Δu+a|u| p－1 u－b|u| q－1 u =0整体弱解的存在性 [ 3 3 ] ，卞春雨和白云成将上述方程推广到含有弱项阻尼项 au t 的高阶波动方程，并得到了整体弱解的存在惟一性 [ 4 4 ] .另外沈继红证明了含有强阻尼项 aΔu t 高维广义Boussmesq方程整体解的存在性与不存在性的门槛结果 [ 5 5 ] .文献 [ 6 6 ] 针对非线性阻尼进行了讨论. 最近刘亚成 [ 7 7 ] 给出了如下含有多个非线性源项的波动方程 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465156&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465156&type=middle 其中 a i > 0,p i 满足：1 ＜ p 1 ＜p 2 ＜…＜p l ＜∞ n =1,2或 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465162&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465162&type=middle ，作者利用位势井理论并结合凸性方法得到了整体解的存在惟一性，并对解的真空隔离现象进行了分析.2011年徐润章 [ 8 8 ] 将上述方程推广至如下具有组合型非线性源项的波动方程 u tt －u xx +u xxxx +f(u) xx =0，其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465165&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465165&type=middle 或 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465168&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465168&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465168&type=middle 而且常数 a i ,b j > 0,参数 p i ,q j 满足类似公式(4)的条件，作者利用一种新的分析技巧，研究了整体解的存在惟一性与爆破现象. 需要指出的是上述大部分结论都没有考虑阻尼项对波动方程的影响，虽然文献 [ 4 4 ] 和 [ 5 5 ] 研究两种不同阻尼形式对波动方程的影响，但他们给出的阻尼项都是线性的，当阻尼项具有非线性形式au t m－2 u t 时 [ 6 6 ] ，他们的分析方法将不再适用.目前，对于同时含有非线性阻尼项和多个源项的波动方程(1)~(4)还未有任何结论，最近文献 [ 9 9 ] 证明了一类含有非线性阻尼和单个源项的波动方程整体解的存在性，本文将文献 [ 7 7 ~ 9 9 7~9 ] 结论推广到含有多个源项的情况，并分析同时含有非线性阻尼与多个源项的波动方程的整体解的存在性与渐近稳定性. 主要引理 1 为了计算方便，引入下面记号 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4464998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4464998&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4464998&type=middle 为了得到文本的结果，引入下面范数 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465004&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465004&type=middle 和集合 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465172&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465172&type=middle 定义1 若 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465174&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465174&type=middle 并且满足： ( i ) 对任意函数 v∈H 0 2 (Ω)∩L m (Ω) 和 a.e .0≤ t ＜T ，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465011&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465011&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465011&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465015&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465015&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465015&type=middle (ii) u(x,0)=u 0 (x) 在 H 0 2 (Ω)∩L m (Ω) 中， u t (x,0)=u 1 (x) 在 H 0 1 (Ω)∩L m (Ω) 中. 则称 u(x,t) 为方程(1)~(3)的弱解. 记 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465177&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465177&type=middle 为空间 H 0 2 (Ω)∩L m (Ω) 的基函数系，利用Galerkin方法构造方程(1)~(4)的近似解 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465019&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465019&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465019&type=middle 其中 d l s (t) 满足如下常微分方程系统： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465022&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465022&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465179&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465179&type=middle ,在 H 0 2 (Ω)∩L m (Ω) 中， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465182&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465182&type=middle ,在 H 0 1 (Ω)∩L m (Ω) 中. 将方程(6)两边同乘以d l s (t)′并求和可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465025&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465025&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465025&type=middle 为了研究方程(1)整体弱解的存在性，下面给出本文的主要引理. 引理1 [ 10 10 ] (Sobolev嵌入定理)设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465185&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465185&type=middle 有界且有锥性质，如果m p ＜n ，有 H m,p (Ω) 嵌入 L q (Ω) ，其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465187&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465187&type=middle ；如果 mp=n ，有 H m,p (Ω) 嵌入 L q (Ω ),其中 p≤q＜∞ . 引理2 [ 9 9 ] 设函数 u(x,t)∈L ∞ 0,T;H 0 2 (Ω)，u t (x,t)∈L ∞ (0,T;H 0 1 (Ω))∩L m (Q T )，u(x,0)=u 0 (x)，u t (x,0)=u 1 (x) ，如果 u tt －Δu－Δu t +Δ 2 u－Δu tt =h ，其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465033&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465033&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465033&type=middle 那么对任意0 ＜ t ＜T 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465040&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465040&type=middle 引理3 [ 9 9 ] 如果函数列 u it 满足: ( i ) u it →u t 在 L m Q T 中弱收敛. (ii) a|u it | m－2 u it →ξ 在 L m′ Q T 中弱收敛，其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465190&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465190&type=middle (iii) 对任意0 ＜t＜T ，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465192&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465192&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465192&type=middle 则 ξ=a|u t | m－2 u t . 整体弱解的存在性 1 在这一部分，主要讨论方程(1)~(4)的整体弱解的存在性. 定理1 设 u 0 (x)∈H 0 2 (Ω)∩L m (Ω)，u 1 (x)∈H 0 1 (Ω)∩L m (Ω) ，则对任意T > 0，方程(1)~(4)存在一个整体弱解 u(x,t) ,且 u t (x,t)∈L ∞ 0,T;H 0 1 (Ω))∩L m (Q T )，u(x,t)∈L ∞ 0,T;H 0 2 (Ω)∩L p i (Ω) 和 u(x,t)∈L ∞ 0,T;H 0 2 (Ω)∩L q j (Ω) . 证将方程(7)两边积分可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465047&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465047&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465047&type=middle 利用 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465193&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465193&type=middle 和Minkowskal不等式可得： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465054&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465054&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465054&type=middle 再利用Holder不等式得： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465060&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465060&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465060&type=middle 由Young不等式可知： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465195&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465195&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465195&type=middle 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465197&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465197&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465200&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465200&type=middle 所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465064&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465064&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465064&type=middle 于是经过计算后得： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465070&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465070&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465070&type=middle 选择ε使得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465202&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465202&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465202&type=middle 可得： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465075&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465075&type=middle 利用公式(4)以及方程(6)的初始条件可知，M(T)是仅与T有关的常数，从而 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465204&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465204&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465206&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465206&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465206&type=middle 都是有限的，它们被仅与 T 有关的常数控制，因此 u l ∈L ∞ 0,T;H 0 2 Ω∩L p i Ω，u l ∈L ∞ (0,T;H 0 2 (Ω)∩L q j (Ω)) 和 u lt ∈L ∞ (0,T;H 0 1 (Ω)∩L m (Q T )) . 由于 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465208&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465208&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465210&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465210&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465210&type=middle 所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465080&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465080&type=middle 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465212&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465212&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465214&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465214&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465214&type=middle 于是可以选取 {u l } 的子数列{ u v }，当 u v →∞时，可得： u v →u 在 L ∞ 0,T;H 0 2 Ω∩L p i Ω 和 L ∞ 0,T;H 0 2 Ω∩L q j Ω 上弱*收敛， u v →u 在 Q T =Ω×[0,T) 上几乎处处收敛， u vt →u t 在 L ∞ 0,T;H 0 1 Ω 上弱*收敛， u vt →u t 在 L m (Q T ) 上弱收敛， a|u vt | m－2 u vt →ξ 在 L m′ (Q T )上弱收敛. 利用文献 [ 11 11 ] 中的Lions引理可知，对任意 i =1,2,…, l 1 和 j =1,2,…, l 2 有 a i |u v | p i －2 u v →a i u p i －2 u 在 L ∞ 0,T;L p′ i Ω上弱*收敛， b j |u v | q j －2 u v →b j u q j －2 u 在 L ∞ 0,T;L q′ j Ω 上弱*收敛. 对(6)式先从0到 t 积分，然后取 v →∞，再对 t 微分可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465084&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465084&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465084&type=middle 选取 v(x)∈H 0 2 (Ω)∩L m (Ω) ，由 w s (x) 在 H 0 2 (Ω)∩L m (Ω) 中稠密可得 v(x)=∑∞s=1d s w s (x) ,其中 d s 为给定的函数，所以对任意 v(x)∈H 0 2 (Ω)∩L m (Ω) 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465088&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465088&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465088&type=middle 下证 ξ=a|u t | m－2 u t ，记 g(s)=a|s m－2 |s ，从(6)式可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465093&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465093&type=middle 令 v →∞得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465098&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465098&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465098&type=middle 利用引理2中的公式(8)可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465103&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465103&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465103&type=middle 由条件(4)可得m ′ ＜p′ l 1 ＜…＜p′ 1 ＜q′ 1 ＜…＜q′ l 2 ，于是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465107&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465107&type=middle 所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465216&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465216&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465216&type=middle ，从引理2知 h∈L 2 (0,T;H －1 (Ω)+L m′ (Q T )) ，可取 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465111&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465111&type=middle 使得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465218&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465218&type=middle 由引理3可得 ξ=a|u t | m－2 u t .于是由公式(9)可得方程(1)~(4)存在一个整体弱解. 弱解的渐近稳定性 1 为了得到整体弱解的渐近稳定性，首先给出下面引理. 引理4 [ 9 9 ] 如果 u(x,t) 为方程(1)~(4)的解，则能量泛函 E(t) 满足： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465115&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465115&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465115&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465119&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465119&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465119&type=middle 引理5 设 (u 0 ,u 1 )∈W×H 0 1 Ω ，在条件(4)下，如果 b j =0(j=1,2,…,l 2 ) 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465123&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465123&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465123&type=middle 其中 C *p i 是 H 0 1 (Ω) 嵌入到 L p i (Ω) 的Sobolev常数，则方程(1)~(3)的整体弱解 u(t)∈W . 证 由于 I(u 0 ) > 0且 I(ut) 关于时间连续，所以存在 t 1 > 0，使得对任意0≤ t≤t 1 ，有 I(u(t)) > 0，于是当 b j =0时有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465127&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465127&type=middle 所以对任意 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465219&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465219&type=middle 利用引理4可知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465130&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465130&type=middle 如果 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465220&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465220&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465220&type=middle ，利用 u∈H 0 2 (Ω) 可得 u =0，从而 u∈W ； 如果 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465221&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465221&type=middle ，利用引理1和公式(12)可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465132&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465132&type=middle 其中 C *p i 为 H 0 1 (Ω) 到 L p i (Ω) 的Sobolev常数，即对任意 0≤t≤t 1 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465222&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465222&type=middle 所以 u∈W ，重复上面的步骤可知对任意0≤ t ＜T ，有 u(t)∈W . 引理6 在引理5的条件下，方程(1)~(3)的整体弱解满足 ‖u(t)‖ m m ≤LE(t) ,其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465135&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465135&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465135&type=middle 证 仿照引理5的证明过程可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465223&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465223&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465223&type=middle 引理7 在引理5的条件下，设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465138&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465138&type=middle 则对任意 u∈W ，存在常数 C 1 ，C 2 > 0，使得当ε足够小时有C 1 E(t)≤F(t)≤C 2 E(t) . 证由于 p i 满足条件(4)，所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465224&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465224&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465224&type=middle ，利用文献 [ 9 9 ] 类似的估计方法即可得到结论. 定理2 设 (u 0 ,u 1 )∈W×H 0 1 (Ω) ，在引理5的条件下，存在常数 M,k>0 ,使得方程(1)~(4)的整体弱解满足 E(t)≤M e －kt (t≥0) . 证 由 F(t) 的定义得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465225&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465225&type=middle ，利用方程(1)和引理4，并结合Poincare不等式可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465142&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465142&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465142&type=middle 对于常数 λ > 0，利用能量泛函 E(t) 的定义以及公式(14)可知： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465145&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4465145&type=middle 最后结合引理6并使用文献 [ 9 9 ] 类似的方法可得存在常数 k > 0，使得F′(t)≤kF(t)，由 Gronwall 等式可知 F(t)≤F(0) e － kt ，由引理7可知，存在常数 M > 0，使得： E(t)≤M e － kt ( t > 0). 参考文献 SATTINGER D H. On global solution of nonlinear hyperbolic equations[J]. Archive for Rational Mechanics and Analysis,1968,30:148-172. PAYNE L E, SATTINGER D H. Saddle points and instability of nonlinear wave equations[J]. Israel Journal of Mathematics,1975,22(3-4):273-303. 叶朝辉,罗显康. 具有两个异号非线性源项的波动方程的整体解[J]. 西南民族大学学报(自然科学版),2007,33(4):718-721. YE C H, LUO X K.Existence of global solutions for wave equations with two nonlinear source terms of different signs[J]. Journal of Southwest University for Nationalities (Natural Science Edition), 2007,33(4):718-721(Ch). 卞春雨,白玉成. 一类具有非线性异号源项波动方程的初边值问题[J]. 哈尔滨师范大学自然科学学报,2010,26(2):32-34. BIAN C Y, BAI Y C.Initial boundary value problem for wave equations with nonlinear source terms of different signs[J]. Natural Sciences Journal of Harbin Normal University, 2010,26(2):32-34(Ch). 沈继红,张明有,杨延冰,等. 具阻尼的高维广义Boussinesq方程的Cauchy问题的整体适定性[J]. 数学物理学报,2014,34A(5):1173-1187. SHEN J H, ZHANG M Y, YANG Y B, et al . Global well-posedness of cauchy problem for damped multidimensional generalized boussinesq equations[J]. Acta Mathematica Scientia, 2014,34A(5):1173-1187(Ch). MESSAOUDI S A. Blow up of positive-initial-energy solutions of nonlinear viscoelastic hyperbolic equation[J]. J Math Anal Appl, 2006,320:902-915. 刘亚成,徐润章,于涛. 具有多个非线性源项的波动方程[J]. 应用数学和力学,2007,28(9):1079-1086. LIU Y C, XU R Z, YU T.Wave equations with several nonlinear source terms[J]. Applied Mathematics and Mechanics, 2007,28(9):1079-1086(Ch). XU R Z, XU C, YU T. Well-posedness of nonlinear wave equation with combined power-type nonlinearities[J]. Mathematical Methods in the Applied Sciences,2011,34(8):869-895. DI H F, SHANG Y D. Global existence and asymptotic behavior of solutions for the double dispersive-dissipative wave equation with nonlinear damping and source terms [DB/OL]. [2015-02-12] . http://paperity.org/p/61648786/global-existence-and-asymptotic-behavior-of-solutions-for-the-double-dispersive DOI 10.1186/s13661-015-0288-6. http://paperity.org/p/61648786/global-existence-and-asymptotic-behavior-of-solutions-for-the-double-dispersive DOI 10.1186/s13661-015-0288-6. 李开泰,马逸尘,王立周. 广义函数和Sobolev空间[M]. 西安：西安交通大学出版社,2008: 99-100. LI K T, MA Y C, WANG L Z.Generalized Function and Sobolev Space[M]. Xi’an: Xi’an Jiao Tong University press, 2008: 99-100(Ch). LIONS J L. Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires[M]. Paris ：Dounod Gauthier-Villars, 1969：4-27. 10.14188/j.1671-8836.2016.05.010 O 175.2 whdxxblxb-62-5-464 安徽省自然科学基金面上项目(1508085MA10)；宿州学院优秀青年人才支持计划重点项目(2016XQNRL003)；宿州学院科研平台开放课题(2014YKF30);国家大学生创新训练计划项目(201410379021) 2015-09-09 2016-10-24 2020-12-10 版权所有 © 《武汉大学学报(理学版)》编辑部 Copyright © 2016 Journal of Wuhan University (Natural Science Edition). All rights reserved. 2016 武汉大学学报(理学版) 5