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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 $$A(t)\vert u_t\vert ^{m-2}u_t-\Delta u=\vert u\vert ^{p-2}u,\quad x\in\Omega,\quad t\in J,$$ $$u(x,t)=0,\quad x\in\partial\Omega,\quad t\in J,$$ $$u(x,0)=u_0,\quad x\in\Omega,$$ where $J=[0,\infty)$ and $\Omega$ is a bounded open subset of $\Bbb R^n$, $u:\Omega\times J\to \Bbb R^N$, $N\ge1$, is studied. Furthermore $A$ is assumed to be of class $C(J;\Bbb R^N\times \Bbb R^N)$ and $$\langle A(t)v,v\rangle\ge c_0\vert v\vert ^2\quad\text{for all $t\in J$ and }v\in \Bbb R^N,$$ where $\langle\cdot\,,\cdot\rangle$ is the inner product in $\Bbb R^N$ and $c_0>0$. Let $2<p\le2n/(n-2)$ when $n\ge3$, while $p>2$ when $n\in\{1,\,2\}$. Suppose that $u_0\in[H^1_0(\Omega)]^N$, $\Vert \nabla u\Vert ^2_2\ge\Vert u\Vert _p^p$, and $$C^p\left[\frac{2p}{p-2}\,\left(\frac12\Vert \nabla u\Vert ^2_2-\frac1p\Vert u\Vert _p^p\right)\right]^{(p-2)/2}<1,$$ where $C=C(n,q,\Omega)$ is the best constant of the embedding $H^1_0(\Omega)\hookrightarrow L^q(\Omega)$. The authors prove that then the energy of the solution $u$ decays exponentially if $m=2$, and polynomial when $m>2$. For the entire collection see [Zbl 1068.35001].

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