双参数非线性非局部奇摄动抛物型初始-边值问题的广义解

研究了一类广义抛物型方程奇摄动问题.首先在一定的条件下, 提出了一类具有两参数的非线性非局部广义抛物型方程初始 边值问题.其次证明了相应问题解的存在性.然后, 通过Fredholm积分方程得到了初始 边值问题的外部解.再利用泛函分析理论和伸长变量及多重尺度法, 分别构造了初始 边值问题广义解的边界层、初始层项，从而得到了问题的形式渐近展开式.最后利用不动点理论证明了对应的非线性非局部广义抛物型方程的奇异摄动初始 边值问题的广义解的渐近展开式的一致有效性.

Generalized Solutions to Nonlinear Nonlocal Singularly Perturbed Parabolic Initial-Boundary Problems With Two Parameters

A class of generalized parabolic equation singular perturbation problems were considered. Firstly, under suitable conditions, a class of nonlinear nonlocal generalized parabolic equation initial-boundary value problems with two parameters were raised. Secondly, the existence of solutions to corresponding problems was proved. Next, from the Fredholm integral equation, the outer solutions to the initial-boundary value problems were found, and the boundary and initial layer terms were structured by means of the theory of functional analysis, the stretched variables and the multiscale methods, respectively. Then the formal asymptotic expansion of the problem was obtained. Finally, according to the fixed point theorem, the uniform validity of the asymptotic expansion of generalized solutions to the corresponding nonlinear nonlocal initial-boundary value problems was proved.