一类伪抛物方程的非线性扰动

Nonlinear Perturbation for a Class of Pseudoparabolic Equations

非线性伪抛物方程和一些重要的物理过程有着密切的关系,研究了一类伪抛物方程Δu+ut-ut-f(x,t,u)=F x,t,u,xui初边值问题的非线性扰动问题。首先在Hilbert空间中建立了强制不等式,利用同胚方法和抽象的反函数定理,得到了半线性伪抛物方程初边值问题解的存在性和惟一性定理。在此基础上,讨论了对应的非线性扰动。通过构造相应的紧算子,利用同伦对算子进行估计并利用Schauder不动点定理,给出了非线性扰动问题解的存在定理。

This paper discusses the nonlinear perturbation of the initial-boundary value problems for a class of nonlinear pseudoparabolic equations △（u＋δu/δt）-δu/δt-f（x,tu）=F（x,t,u,δu/δxi）A coercivity inequality in Hilbert space is founded. By using homeomorphism method and the extended inverse function theorem, the existence and uniqueness of the solution for the semi-linear pseudoparabolic equations is obtained. Based on this, the relevant nonlinear perturbation of the problems is proposed. Through constructing a corresponding compact operator, estimating the operator with homotopic method and using Schauder fix theorem, the existence of the solution of the perturbation problems is given.