Nonlinear Method of Angular Boundary Functions Under the Influence of the Inflection Point

In the rectangle Ω = {(x, t) | 0 < x < 1, 0 < t < T} the initial boundary value problem for the singularly perturbed parabolic equation ε2 (︂ a2 ∂2u ∂x2 − ∂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. is considered. It is assumed that at the angular points (k, 0) of the rectangle Ω, where k = 0 or 1, the function F(u) = F(u, k, 0, 0) takes the form F(u) = u3 − u30 , где u0 = u0(k) < 0. The nonlinear method of angular boundary functions is used to construct the asymptotics of the solution to the problem. Earlier, we considered the case when the boundary value of φ at the angular points is separated from the inflection point u = 0 by the condition u0(k) < φ(k) ≤ u0(k) 2 < 0, at which functions of the “simplest” form suitable in the entire domain in question fitted to the role of barrier functions. In this work, the case u0(k) 2 < φ(k) < 0 is considered, where the domain has to be divided into parts, the barrier functions have to be constructed in each subdomain taking into account their continuous junction at the common boundaries of the subdomains, and then the piecewise continuous lower and upper solutions have to be smoothed. As a result, a complete asymptotic expansion of the solution when ε → 0 is obtained and its uniformity in the closed rectangle is justified.