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Existence of global attractors in \(L^p\) for \(m\)-Laplacian parabolic equation in \(\mathbb{R}^N\). (English) Zbl 1183.35050
Summary: We study the long-time behavior of solution for the \(m\)-Laplacian equation \[ u_t- \text{div}(|\nabla u|^{m-2}\nabla u)+ \lambda|u|^{m-2}u+ f(x,u)= g(x) \quad\text{in }\mathbb R^N\times\mathbb R^+, \] in which the nonlinear term \(f(x,u)\) is a function like \(f(x,u)= -h(x)|u|^{q-2}u\) with \(h(x)\geq0\), \(2\leq q<m\), or \(f(x,u)= a(x)|u|^{\alpha-2}u- h(x)|u|^{\beta-2}u\) with \(a(x)\geq h(x)\geq0\) and \(\alpha>\beta\geq m\). We prove the existence of a global \((L^2(\mathbb R^N),L^p(\mathbb R^N))\)-attractor for any \(p>m\).

MSC:
35B41 Attractors
35K65 Degenerate parabolic equations
35K59 Quasilinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
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References:
[1] doi:10.1016/S0167-2789(98)00304-2 · Zbl 0953.35022
[2] doi:10.1016/j.jmaa.2005.05.003 · Zbl 1090.35040
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