Numerical verification for existence of a global-in-time solution to semilinear parabolic equations

Abstract

This paper presents a method of numerical verification for the existence of a global-in-time solution to a class of semilinear parabolic equations. Such a method is based on two main theorems in this paper. One theorem gives a sufficient condition for proving the existence of a solution to the semilinear parabolic equations with the initial point $t=t^{\prime }\ge 0$. If the sufficient condition does not hold, the other theorem is used for enclosing the solution for time $t\in (0,\tau ],\tau >0$ in a neighborhood of a numerical solution. Numerical results of obtaining a global-in-time solution for a certain semilinear parabolic equation are also given.