Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem

Abstract

The authors consider the problem ut−div(|∇u|p−2∇u)=0 in (0,∞)×RN, u(x,0)=u0(x). They show that if N≥2 and 1<p<2N/(N+1) the solution has a finite extinction time for each u0∈Lm, m=N(2/p−1), and if N=1, p>1 or N≥2, p≥2N/(N+1) then conservation of total mass holds, i.e., ∫u(t,x)dx=∫u0(x)dx. Moreover the regularizing and decay estimate for ∥u(t)∥m (1<m≤∞) is proved for u0∈Lm0 with m0≥1, which is the extension of the corresponding result for bounded domains by L. Véron [same journal (5) 1 (1979), no. 2, 171–200] to the case of whole space. Finally the finite extinction time problem is discussed for the problem in a bounded domain, extending the result by A. Bamberger [J. Funct. Anal. 24 (1977), no. 2, 148–155].