Numerical solution of a Cauchy problem for nonlinear reaction diffusion processes

In this paper, the authors propose a numerical method to compute the solution of the Cauchy problem: wt-(wmwx)x=wpwt-(wmwx)x=wp, the initial condition is a nonnegative function with compact support, m>0m>0, p⩾m+1p⩾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⩽pm+3p>m+3, if the initial condition is sufficiently small, a global numerical solution exists, and if p⩾m+3p⩾m+3, for large initial condition, the solution is unbounded.