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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
\[ A(t)| u_t| ^{m-2}u_t-\Delta u=| u| ^{p-2}u,\quad x\in\Omega,\quad t\in J, \]
\[ u(x,t)=0,\quad x\in\partial\Omega,\quad t\in J, \]
\[ u(x,0)=u_0,\quad x\in\Omega, \]
where \(J=[0,\infty)\) and \(\Omega\) is a bounded open subset of \(\mathbb R^n\), \(u:\Omega\times J\to \mathbb R^N\), \(N\geq1\), is studied. Furthermore \(A\) is assumed to be of class \(C(J;\mathbb R^N\times \mathbb R^N)\) and \[ \langle A(t)v,v\rangle\geq c_0| v| ^2\quad\text{for all \(t\in J\) and }v\in \mathbb R^N, \] where \(\langle\cdot\,,\cdot\rangle\) is the inner product in \(\mathbb R^N\) and \(c_0>0\). Let \(2<p\leq2n/(n-2)\) when \(n\geq3\), while \(p>2\) when \(n\in\{1,\,2\}\). Suppose that \(u_0\in[H^1_0(\Omega)]^N\), \(\| \nabla u\| ^2_2\geq\| u\| _p^p\), and \[ C^p\left[\frac{2p}{p-2}\,\left(\frac12\| \nabla u\| ^2_2-\frac1p\| u\| _p^p\right)\right]^{(p-2)/2}<1, \] 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].

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K65 Degenerate parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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