On the Approximation of Blow-up Time for Solutions of Nonlinear Parabolic Equations

Abstract

There are many nonlinear parabolic equations whose solutions develop singularity in finite time, say T. In many cases, a certain norm of the solution tends to infinity as time t approaches T. Such a phenomenon is called blow-up, and T is called the blow-up time. This paper is concerned with approximation of blow-up phenomena in nonlinear parabolic equations. For numerical computations or for other reasons, we often have to deal with approximate equations. But it is usually not at all clear if such wild phenomena as blow-up can be well reflected in the approximate equations. In this paper we present rather simple but general sufficient conditions which guarantee that the blow-up time for the original equation is well approximated by that for approximate equations. We will then apply our result to various examples.