Nonlinear Method of Corner Boundary Functions with the Influence of an Inflection Point

In a rectangle $$\Omega = \{ (x,t)\,|\,0 < x < 1,\;0 < t < T\} $$ , we consider an initial-boundary value problem for a singularly perturbed parabolic equation $${{\varepsilon }^{2}}\left( {{{a}^{2}}\frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}} - \frac{{\partial u}}{{\partial t}}} \right) = F(u,x,t,\varepsilon ),\quad (x,t) \in \Omega ,$$ $$u(x,0,\varepsilon ) = \varphi (x),\quad 0 \leqslant x \leqslant 1,\quad u(0,t,\varepsilon ) = {{\psi }_{1}}(t),\quad u(1,t,\varepsilon ) = {{\psi }_{2}}(t),\quad 0 \leqslant t \leqslant T.$$ It is assumed that, at the corner points $$(k,0)$$ of the rectangle $$\Omega $$ , where $$k = 0$$ or 1, the function $$F(u) = F(u,k,0,0)$$ has the form $$F(u) = {{u}^{3}} - u_{0}^{3},\quad {\text{where}}\quad {{u}_{0}} = {{u}_{0}}(k) < 0.$$ To construct the asymptotics of the solution, we use a nonlinear method of corner boundary functions. Previously, the case was considered when the boundary value $$\varphi $$ at the corner points is separated from the inflection point $$u = 0$$ by the condition $${{u}_{0}}(k) < \varphi (k) \leqslant \frac{{{{u}_{0}}(k)}}{2} < 0,$$ in which the role of barrier functions was played by functions of the simplest type, suitable in the entire domain. In this paper, we consider the case of $$\frac{{{{u}_{0}}(k)}}{2} < \varphi (k) < 0,$$ in which the domain has to be divided into parts in order to construct in each subdomain its individual barrier functions taking into account their continuous matching at the common boundaries of the subdomains and then to smooth out the piecewise continuous lower and upper solutions. As a result, we obtain a complete asymptotic expansion of the solution at $$\varepsilon \to 0$$ and substantiate its uniformity in the closed rectangle.