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Stabilization of solutions of an anisotropic quasilinear parabolic equation in unbounded domains. (English. Russian original) Zbl 1304.35355

Proc. Steklov Inst. Math. 278, 106-120 (2012); translation from Tr. Mat. Inst. Steklova 278, 114-128 (2012).
Summary: The first initial-boundary value problem with the homogeneous Dirichlet boundary condition and a compactly supported initial function is considered for a model second-order anisotropic parabolic equation in a cylindrical domain \(D = (0,\infty) \times \Omega\). We find an upper bound that characterizes the dependence of the decay rate of solutions as \(t \to \infty\) on the geometry of the unbounded domain \(\Omega\subset \mathbb R^n\), \(n \geq 3\), and on nonlinearity exponents. We also obtain an estimate for the admissible decay rate of nonnegative solutions in unbounded domains; this estimate shows that the upper bound is sharp.

MSC:

35K59 Quasilinear parabolic equations
35B35 Stability in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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