非线性反应扩散问题的广义渐近解

非线性反应扩散问题的广义渐近解 The Generalized Asymptotic Solution for Nonlinear Reaction Diffusion Problem 1 first-author 史娟荣, 女，副教授，主要从事应用微分方程等方面研究. E-mail: ahjdshjr@126.com 史娟荣, 女，副教授，主要从事应用微分方程等方面研究. E-mail: ahjdshjr@126.com ahjdshjr@126.com 2 3 corresp 莫嘉琪, E-mail:mojiaqi@mail.ahnu.edu.cn 莫嘉琪, E-mail:mojiaqi@mail.ahnu.edu.cn mojiaqi@mail.ahnu.edu.cn 安徽机电技术学院, 安徽 芜湖 241002 安徽机电技术学院, 安徽 芜湖 241002 Anhui Technical College of Mechanical and Electrical Engineering, Wuhu 241002, Anhui, China Anhui Technical College of Mechanical and Electrical Engineering, Wuhu 241002, Anhui, China 上海交通大学 数学系, 上海 200240 上海交通大学 数学系, 上海 200240 Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China 安徽师范大学 数学系, 安徽 芜湖 241003 安徽师范大学 数学系, 安徽 芜湖 241003 Department of Mathematics, Anhui Normal University, Wuhu 241003, Anhui, China Department of Mathematics, Anhui Normal University, Wuhu 241003, Anhui, China 研究了具有内部激波层、初始层的非线性反应扩散方程奇摄动广义初始边值问题.在适当的条件下, 讨论了问题的广义外部解.引入伸长变量, 构造了原问题具有内部激波和初始层校正项的广义解的形式渐近展开式.利用泛函分析不动点理论得到了原初始边值问题广义解的存在性和一致有效的内部激波广义渐近解. The nonlinear singular perturbation interior layer and initial layer to the initial boundary value problem for the reaction diffusion equation is considered. Under suitable conditions, the generalized outer solution is discussed. By introducing a stretched variable, the formal asymptotic expansion for the generalized solution with interior layer and initial layer corrective terms is constructed. Finally, using the fixed point theory of functional analysis, the existence and uniformly valid interior shock asymptotic solution for the original initial boundary value problems are proved. 非线性 激波 渐近解 nonlinear shock asymptotic solution 引言 1 非线性激波问题是当前在数学界十分重视的一个研究课题 [ 1 1 , 2 2 ] .近十年来许多学者的研究中都涉及到这一方面的工作.例如，Martinez和Wolanski [ 3 3 ] Kellogg和Kopteva [ 4 4 ] ，Tian和Zhu [ 5 5 ] , Skrynnikov [ 6 6 ] 以及Samusenko [ 7 7 ] 等.本文作者也讨论过了一类奇摄动非线性激波、孤立子、冲击层等问题 [ 8 8 ~ 22 22 8~22 ] , 本文讨论一类广义反应扩散奇摄动激波问题. 考虑如下反应扩散初始-边值问题： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457767&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457767&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457772&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457772&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457777&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457777&type=middle 其中 ε 为正的小参数, T 0 为一个足够大的正常数, 假设: [H 1 ] f ( t , x , u ) 在 ( t , x , u )∈(0, T 0 ]×((-1, x 0 )∪( x 0 , 1))× R 上, g ( t , x ) 在 ( t , x )∈(0, T 0 ]×(-11) 上和 h ( x ) 在 x ∈(-11) 上为充分光滑的函数, 且 g (0, x )= h ( x ); [H 2 ]存在常数 δ > 0, 使得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457782&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457782&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457782&type=middle 代替问题 (1)~(3)，我们来讨论如下广义反应扩散奇摄动初始-边值问题： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457787&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457787&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457787&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457790&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457790&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457790&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457794&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457794&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457794&type=middle 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457796&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457796&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457796&type=middle 而 C 0 ∞ 为 C ∞ ((0, T 0 )×(-1, 1)) 的紧支子集, 表示式 B 1 [ v , u ]为关于 v , u 的双线性算子，( u , v ) 是被定义在 H 1 ((0, T 0 )×(-1, 1))) 中的内积. u , v 是在 (0, T 0 )×(-1, 1)) 中有界并被定义在Sobolev泛函空间 H 1 ((0, T 0 )×(-1, 1))) 上, 并具有有限模： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457799&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457799&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457799&type=middle 我们还假设: [H 3 ]退化问题 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457803&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457803&type=middle 有一个广义解 U 0 ( t , x ), 但在 x 0 ∈(-1, 1) 处 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457805&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457805&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457805&type=middle 外部解 1 设问题 (4)~(6) 的广义外部解 U 为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457807&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457807&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457807&type=middle 将 (7) 式代入 (4), (5) 式, 按 ε 展开非线性项, 合并 ε 的同次幂项, 并令其系数为零.对于 ε i , i =0, 1, …, 可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457809&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457809&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457811&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457811&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457813&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457813&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457815&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457815&type=middle 其中 F i ( i =1, 2, …) 为逐次已知的函数, 其结构从略. 由假设[H 3 ], 广义初值问题 (8), (9) 存在广义解 U 0 ( t , x ), 再由线性问题 (10), (11), 可依次得到广义解 U i ( t , x ), i =1, 2, ….将它们代入 (7) 式, 便得到问题 (4)-(6) 的广义外部解 U .但外部解 U 在 ( t , x 0 )∈(0, T 0 )×(-1, 1) 处未必连续, 而且未必满足条件 (6).为此尚需在 ( t , x 0 )∈(0, T 0 )×(-1, 1) 附近构造广义内部激波层校正项 V 和在 t =0附近构造广义初始层校正项 W . 广义内部激波层校正项 1 我们首先在 ( t , x 0 )∈(0, T 0 )×(-1, 1) 附近构造广义内部激波校正项.引入伸长变量 [ 1 1 , 2 2 ] : σ =| x - x 0 | ε , 这时由广义初始边值问题 (4)-(6) 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457816&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457816&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457816&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457817&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457817&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457817&type=middle 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457818&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457818&type=middle 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457819&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457819&type=middle 这里 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457821&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457821&type=middle 将 (14), (15) 式代入 (12), (13) 式, 按 ε 展开非线性项, 合并 ε 的同次幂项, 并令其系数为零.对于 ε i , i =0, 1, …, 可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457823&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457823&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457823&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457824&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457824&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457827&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457827&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457827&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457830&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457830&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457830&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457833&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457833&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457836&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457836&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457836&type=middle 其中 F i ( i =1, 2, …) 为逐次已知的函数, 其结构也从略.上面和以下出现负下标的项均设为零. 由 (16)~(18) 式和 (19)~(21) 式能依次地得到 v i ( t , σ )( i =0, 1, …), 并且由假设知 v i ( t , σ ) 具有内部激波层性态 [ 1 1 , 2 2 ] : http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457839&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457839&type=middle 这里 k i ( i =0, 1, …) 为正常数. 设 v i = ψ ( ρ ) v i ( ρ =| x - x 0 |), ψ ( ρ ) 为[-1, 1]上充分光滑的函数, 且满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457843&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457843&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457843&type=middle 其中 ρ 0 为适当大的正常数.不失一般性, 为了方便起见, 以下我们仍以 v i 代替 v i . 将 v i ( i =0, 1, …) 代入 (15) 式, 便得到奇摄动反应扩散问题 (4)-(6) 广义解 u 的内部激波层校正项 V . 广义初始层校正项 1 作伸长变量的变换: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457848&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457848&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457848&type=middle 并设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457851&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457851&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457851&type=middle 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457854&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457854&type=middle 考虑到 (23)，将 (24), (25) 式代入 (4)-(6) 式，并把各项按 ε 的幂进行展开.取 ε i ( i =0, 1, 2) 的系数的代数和为零.可得: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457856&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457856&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457859&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457859&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457859&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457863&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457863&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457863&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457867&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457867&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457867&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457871&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457871&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457871&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457875&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457875&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457875&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457879&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457879&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457879&type=middle 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457913&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457913&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457913&type=middle 逐次已知的函数。由 (26)-(28) 式和 (29)-(31) 式能依次地得到 w i ( τ , x )( i =0, 1, …), 并且由假设知 w i ( τ , x ) 具有边界层性态 [ 1 1 , 2 2 ] : http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457883&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457883&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457883&type=middle 这里 k i ( i =0, 1, …) 为正常数. 再令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457886&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457886&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457886&type=middle 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457917&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457917&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457917&type=middle ，并满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457891&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457891&type=middle 其中 r 0 为适当大的正常数.不失一般性, 为了方便起见, 以下仍以 w i 代替 W i . 因此， w i 为具有初始层校正项性态的函数.将所得的结果代入 (25) 式，得到奇摄动反应扩散初始边值问题 (4)-(6) 解 u 的广义初始层校正项 W . 于是我们便构造了反应扩散奇摄动问题 (4)-(6) 广义解u的如下形式渐近展开式: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457896&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457896&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457896&type=middle 广义解的一致有效性 1 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457900&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457900&type=middle 由 (22), (32) 式, 对于充分小的 ε ，使得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457902&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457902&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457902&type=middle 故由假设和泛函分析不动点理论 [ 1 1 , 2 2 ] , 可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457905&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457905&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457905&type=middle 于是有如下定理: 定理1     在假设[H 1 ]-[H 3 ]下, 对于充分小的 ε , 奇摄动反应扩散广义初始边值问题 (4)-(6) 在H 1 ((0, T 0 )×(-1, 1)) 意义下存在一个广义解 u , 且解 u 具有形如 (33) 的一致有效的渐近展开式, 成立 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457908&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457908&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457908&type=middle 其中 U 0 为广义退化问题的解， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457920&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457920&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4457920&type=middle 为奇摄动反应扩散广义初始-边值问题 (4)-(6) 广义外部解的近似值, v i ( i =0, 1, 2, …) 为广义内部激波层校正函数, w i ( i =0, 1, 2, …) 为广义边界层校正函数. 参考文献 DE JAGERE M, JIANG F R. 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