Generic behaviour of one-dimensional blow up patterns

Abstract

This paper concerns the Cauchy problem ut−uxx=up, x∈R, t>0, u(x,0)=u0(x), x∈R, where p>1 and u0(x) is a continuous, nonnegative and bounded function. It has been previously proved that if x=x¯, t=T is a blow-up point, then there are three cases for the asymptotic behavior of a solution near the blow-up point. The main result of this paper is to prove that if u0∈C+0(R), blow-up consists generically of a single point blow-up, with the behavior described in one case (case (b)). Moreover, the behavior is stable under small perturbations in the L∞-norm of the initial value u0.