On quenching with logarithmic singularity

Abstract

In this paper we study the quenching problem for the reaction diffusion equation u t − u xx = f( u) with Cauchy–Dirichlet data, in the case where we have only a logarithmic singularity, i.e., f( u)=ln( αu), α∈(0,1). We show that for sufficiently large domains of x quenching occurs, and that under certain assumptions on the initial function, the set of quenching points is finite. The main result of this paper concerns the asymptotic behavior of the solution in a neighborhood of a quenching point. This result gives the quenching rate for the problem. We also obtain new blow-up results for the equation v t − v xx = αve v − v x 2. These concern the occurrence of blow-up, the blow-up set and the asymptotics in a neighborhood of a blow-up point. The analysis is based on the equivalence between the quenching and the blow-up for these two equations.