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Remark on the Cauchy problem for the evolution \(p\)-Laplacian equation. (English) Zbl 1505.35257

Summary: In this paper, we prove that the semigroup \(S(t)\) generated by the Cauchy problem of the evolution \(p\)-Laplacian equation \(\frac{\partial u}{\partial t}-\operatorname{div}(| \nabla u|^{p-2}\nabla u)=0\) (\(p>2\)) is continuous from a weighted \(L^{\infty}\) space to the continuous space \(C_{0}(\mathbb{R}^{N})\). Then we use this property to reveal the fact that the evolution \(p\)-Laplacian equation generates a chaotic dynamical system on some compact subsets of \(C_{0}(\mathbb{R}^{N})\). For this purpose, we need to establish the propagation estimates and the space-time decay estimates for the solutions first.

MSC:

35K92 Quasilinear parabolic equations with \(p\)-Laplacian
35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35K65 Degenerate parabolic equations
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