Approximation of singularly perturbed parabolic equations in unbounded domains subject to piecewise smooth boundary conditions in the case of solutions that grow at infinity

The initial-boundary value problem in a domain on a straight line that is unbounded in $x$ is considered for a singularly perturbed reaction-diffusion parabolic equation. The higher order derivative in the equation is multiplied by a parameter $\varepsilon^2$, where $\varepsilon\in(0,1]$. The right-hand side of the equation and the initial function grow unboundedly as $x\to\infty$ at a rate of $O(x^2)$. This causes the unbounded growth of the solution at infinity at a rate of $O(\Psi(x))$, where $\Psi(x)=x^2+1$. The initialboundary function is piecewise smooth. When $\varepsilon$ is small, a boundary and interior layers appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristics of the reduced equation passing through the discontinuity points of the initial function. In the problem under examination, the error of the grid solution grows unboundedly in the maximum norm as $x\to\infty$ even for smooth solutions when $\varepsilon$ is fixed. In this paper, the proximity of solutions of the initial-boundary value problem and its grid approximations is considered in the weighted maximum norm $\|\cdot\|^w$ with the weighting function $\Psi^{-1}(x)$; in this norm, the solution of the initial-boundary value problem is $\varepsilon$-uniformly bounded. Using the method of special grids that condense in a neighborhood of the boundary layer or in neighborhoods of the boundary and interior layers, special finite difference schemes are constructed and studied that converge $\varepsilon$-uniformly in the weighted norm. It is shown that the convergence rate considerably depends on the type of nonsmoothness in the initial-boundary conditions. Grid approximations of the Cauchy problem with the right-hand side and the initial function growing as $O(\Psi(x))$ that converge $\varepsilon$-uniformly in the weighted norm are also considered.