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A decay result for a quasilinear parabolic system. (English) Zbl 1082.35029
Bandle, Catherine (ed.) et al., Elliptic and parabolic problems. A special tribute to the work of Haim Brezis. Basel: Birkhäuser (ISBN 3-7643-7249-4/hbk). Progress in Nonlinear Differential Equations and their Applications 63, 43-50 (2005).

The quasilinear parabolic problem

A(t)|u t | m-2 u t -Δu=|u| p-2 u,xΩ,tJ,
u(x,t)=0,xΩ,tJ,
u(x,0)=u 0 ,xΩ,

where J=[0,) and Ω is a bounded open subset of n , u:Ω×J N , N1, is studied. Furthermore A is assumed to be of class C(J; N × N ) and

A(t)v,vc 0 |v| 2 foralltJandv N ,

where ·,· is the inner product in N and c 0 >0. Let 2<p2n/(n-2) when n3, while p>2 when n{1,2}. Suppose that u 0 [H 0 1 (Ω)] N , u 2 2 u p p , and

C p 2p p-21 2u 2 2 -1 pu p p (p-2)/2 <1,

where C=C(n,q,Ω) is the best constant of the embedding H 0 1 (Ω)L q (Ω). The authors prove that then the energy of the solution u decays exponentially if m=2, and polynomial when m>2.

MSC:
35B40Asymptotic behavior of solutions of PDE
35K65Parabolic equations of degenerate type
35K60Nonlinear initial value problems for linear parabolic equations