Numerical solution of a Cauchy problem for nonlinear reaction diffusion processes

Abstract

In this paper, the authors propose a numerical method to compute the solution of the Cauchy problem: w t - ( w m w x ) x = w p , the initial condition is a nonnegative function with compact support, m > 0 , p ⩾ m + 1 . The problem is split into two parts: a hyperbolic term solved by using the Hopf and Lax formula and a parabolic term solved by a backward linearized Euler method in time and a finite element method in space. The convergence of the scheme is obtained. Further, it is proved that if m + 1 ⩽ p < m + 3 , any numerical solution blows up in a finite time as the exact solution, while for p > m + 3 , if the initial condition is sufficiently small, a global numerical solution exists, and if p ⩾ m + 3 , for large initial condition, the solution is unbounded.