Nonlinear method of angular boundary functions in problems with cubic nonlinearities

Abstract

In the rectangle Ω = {(𝑥, 𝑡) | 0 < 𝑥 < 1, 0 < 𝑡 < 𝑇} we consider an initial-boundary value
problem for a singularly perturbed parabolic equation

$$𝜀2(︂𝑎^2((𝜕^2)𝑢/𝜕𝑥^2)−𝜕𝑢/𝜕𝑡)︂= 𝐹(𝑢, 𝑥, 𝑡, 𝜀), (𝑥, 𝑡) ∈ Ω,$$
$$𝑢(𝑥, 0, 𝜀) = 𝜙(𝑥), 0 ⩽ 𝑥 ⩽ 1,$$
$$𝑢(0, 𝑡, 𝜀) = 𝜓1(𝑡), 𝑢(1, 𝑡, 𝜀) = 𝜓2(𝑡), 0 ⩽ 𝑡 ⩽ 𝑇.$$

It is assumed that at the corner points of the rectangle the function 𝐹 with respect to the variable 𝑢 is cubic. To construct the asymptotics of the solution to the problem, the nonlinear method of angular boundary functions is used, which involves the following steps:
1) splitting the area into parts;
2) construction in each subdomain of lower and upper solutions of the problem;
3) continuous joining of the lower and upper solutions on the common boundaries of the subdomains;
4) subsequent smoothing of piecewise continuous lower and upper solutions.
In the present work, we succeeded in constructing barrier functions suitable for the entire region at once. The form of barrier functions is determined using boundary-layer functions that are solutions of ordinary differential equations, as well as taking into account the necessary properties of the desired solutions. As a result, a complete asymptotic expansion of the solution for 𝜀 → 0 is constructed and its uniformity in a closed rectangle is justified.