Nonlinear method of angular boundary functions for singularly perturbed parabolic 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

$$
\varepsilon^2\left(a^2\frac{\partial^2 u}{\partial x^2}-\frac{\partial u}{\partial t}\right)=F(u,x,t,\varepsilon),   \quad  (x,t)\in \Omega,
$$
$$
u(x,0,\varepsilon)=\varphi(x),    \quad 0\le x\le 1,
$$
$$
u(0,t,\varepsilon)=\psi_1(t), \quad u(1,t,\varepsilon)=\psi_2(t), \quad 0\le t\le T.
$$

Research is carried out under the assumption that at the corner points (𝑘, 0) of the rectangle Ω, where 𝑘 = 0 or 1, the function 𝐹(𝑢) = 𝐹(𝑢, 𝑘, 0, 0) is cubic and has the form

$$
F(u)=(u-\alpha(k))(u-\beta(k))(u-\bar u_0(k)), \quad\mbox{где}\quad  \alpha(k)\leq\beta(k)<\bar u_0(k).
$$

The nonlinear method of angular boundary functions is used, which combines the (linear) method of angular boundary functions, the method of upper and lower solutions (barriers), and the method of differential inequalities. Under the condition 𝜙(𝑘) > ¯𝑢0(𝑘), a complete asymptotic expansion of the solution for 𝜀 → 0 is constructed and its uniformity in a closed rectangle is substantiated.
Previously, the following cases of cubic nonlinearities were considered:

$$
F(u)=u^3-\bar u^3_0, \quad\mbox{where}\quad \bar u_0=\bar u_0(k)>0,
$$ 

under the assumption that the boundary value 𝜙(𝑘) > ¯𝑢0(𝑘), as well as the case

$$
F(u)=u^3-\bar u^3_0, \quad\mbox{where}\quad \bar u_0=\bar u_0(k)< 0,
$$

under the assumption that the boundary value 𝜙(𝑘) is contained in the interval 

$$
\bar u_0<\varphi(k)<\frac{\bar u_0}{2}< 0.
$$