Abstract

For a singularly perturbed parabolic equation

\({{\epsilon }^{2}}\left( {{{a}^{2}}\frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}} - \frac{{\partial u}}{{\partial t}}} \right) = F(u,x,t,\epsilon )\)

in a rectangle, a problem with boundary conditions of the first kind is considered. It is assumed that, at the corner points of the rectangle, the function \(F\) is cubic in the variable \(u\). A complete asymptotic expansion of the solution at \(\varepsilon \to 0\) is constructed, and its uniformity in a closed rectangle is substantiated.

中文翻译：

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具有三次非线性奇摄动抛物型方程边值问题的角边界层。

摘要

对于奇摄动抛物方程

\（{{\ epsilon} ^ {2}} \ left（{{{a} ^ {2}} \ frac {{{{\ partial} ^ {2}} u}} {{\ partial {{x} ^ {2}}}}-\ frac {{\ partial u}} {{\ partial t}}} \ right）= F（u，x，t，\ epsilon）\）

在矩形中，考虑第一种边界条件的问题。假设在矩形的拐角处，函数\（F \）在变量\（u \）中是三次的。构造了该解在\（\ varepsilon \ to 0 \）处的一个完整的渐近展开，并证明了其在闭合矩形中的均匀性。