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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

$A\left(t\right)|{u}_{t}{|}^{m-2}{u}_{t}-{\Delta }u={|u|}^{p-2}u,\phantom{\rule{1.em}{0ex}}x\in {\Omega },\phantom{\rule{1.em}{0ex}}t\in J,$
$u\left(x,t\right)=0,\phantom{\rule{1.em}{0ex}}x\in \partial {\Omega },\phantom{\rule{1.em}{0ex}}t\in J,$
$u\left(x,0\right)={u}_{0},\phantom{\rule{1.em}{0ex}}x\in {\Omega },$

where $J=\left[0,\infty \right)$ and ${\Omega }$ is a bounded open subset of ${ℝ}^{n}$, $u:{\Omega }×J\to {ℝ}^{N}$, $N\ge 1$, is studied. Furthermore $A$ is assumed to be of class $C\left(J;{ℝ}^{N}×{ℝ}^{N}\right)$ and

$〈A\left(t\right)v,v〉\ge {c}_{0}{|v|}^{2}\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}t\in J\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}v\in {ℝ}^{N},$

where $〈·\phantom{\rule{0.166667em}{0ex}},·〉$ is the inner product in ${ℝ}^{N}$ and ${c}_{0}>0$. Let $2 when $n\ge 3$, while $p>2$ when $n\in \left\{1,\phantom{\rule{0.166667em}{0ex}}2\right\}$. Suppose that ${u}_{0}\in {\left[{H}_{0}^{1}\left({\Omega }\right)\right]}^{N}$, ${\parallel \nabla u\parallel }_{2}^{2}\ge {\parallel u\parallel }_{p}^{p}$, and

${C}^{p}{\left[\frac{2p}{p-2}\phantom{\rule{0.166667em}{0ex}}\left(\frac{1}{2}{\parallel \nabla u\parallel }_{2}^{2}-\frac{1}{p}{\parallel u\parallel }_{p}^{p}\right)\right]}^{\left(p-2\right)/2}<1,$

where $C=C\left(n,q,{\Omega }\right)$ is the best constant of the embedding ${H}_{0}^{1}\left({\Omega }\right)↪{L}^{q}\left({\Omega }\right)$. The authors prove that then the energy of the solution $u$ decays exponentially if $m=2$, and polynomial when $m>2$.

##### MSC:
 35B40 Asymptotic behavior of solutions of PDE 35K65 Parabolic equations of degenerate type 35K60 Nonlinear initial value problems for linear parabolic equations
##### Keywords:
exponential decay; polynomial decay