On a general approach to extinction and blow-up for quasi-linear heat equations

Abstract

The authors study asymptotic behaviour of positive solutions of equations of the type ut =Δφ(u)±Q(u), where φ′ and Q are given positive functions. By determining an auxiliary function F(u) appearing in an expression posed by A. Friedman and B. McLeod, they obtain asymptotic estimates of solutions as t→T, blow-up or extinction time. These estimates have been established by other authors using different methods. Moreover, the paper poses a conjecture that, if the behaviour of u(0,t) as t→T near a blow-up or extinction point is known, all the information about the corresponding asymptotic expansions on small compact subsets near the origin is encoded in the first order ODE φ′(u)ur+rF(u)=0 for r>0 as t→T, where an optimal choice of F(u) is indicated in the paper.