Journal of Inequalities and Applications 
Volume 3 (1999), Issue 1, Pages 1-23
doi:10.1155/S1025583499000016

Explicit exponential decay bounds in quasilinear parabolic problems

G. A. Philippin1 and S. Vernier Piro2

 1Département de mathématiques et de statistique, Université Laval, Québec G1K 7P4, Canada
2Dipartimento di matematica, Università di Cagliari, Cagliari 09124, Italy
Received 2 March 1997; Revised 3 September 1997

Abstract

This paper deals with classical solutions u(x,t) of some initial boundary value problems involving the quasilinear parabolic equation g(k(t)|∇u|2)Δu+f(u)=ut,  x∈Ω,t>0, where f,g,k are given functions. In the case of one space variable, i.e. when Ω:=(−L,L), we establish a maximum principle for the auxiliary function Φ(x,t):=e2αt{1k(t)∫0k(t)ux2g(ξ)dξ+αu2+2∫0uf(s)ds}, where a is an arbitrary nonnegative parameter. In some cases this maximum principle may be used to derive explicit exponential decay bounds for |u| and |ux|. Some extensions in N space dimensions are indicated. This work may be considered as a continuation of previous works by Payne and Philippin (Mathematical Models and Methods in Applied Sciences, 5 (1995), 95–110; Decay bounds in quasilinear parabolic problems, In: Nonlinear Problems in Applied Mathematics, Ed. by T.S. Angell, L. Pamela, Cook, R.E., SIAM, 1997).