The Fujita exponent for finite difference approximations of nonlocal and local semilinear blow-up problems

TL;DR

This paper investigates finite difference methods for reaction-diffusion equations with nonlocal operators, establishing the discrete Fujita exponent, convergence of blow-up times, and analyzing asymptotic behavior and eigenvalues.

Contribution

It introduces the discrete Fujita critical exponent for nonlocal reaction-diffusion problems and proves convergence of discrete blow-up times to continuous ones.

Findings

Derived the discrete Fujita critical exponent.

Proved convergence of discrete blow-up times.

Analyzed asymptotic behavior and eigenvalue problems.

Abstract

We study monotone finite difference approximations for a broad class of reaction-diffusion problems, incorporating general symmetric L\'evy operators. By employing an adaptive time-stepping discretization, we derive the discrete Fujita critical exponent for these problems. Additionally, under general consistency assumptions, we establish the convergence of discrete blow-up times to their continuous counterparts. As complementary results, we also present the asymptotic-in-time behavior of discrete heat-type equations as well as an extensive analysis of discrete eigenvalue problems.