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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 [4] doi:10.1619/fesi.50.449 · Zbl 1180.35125 [5] doi:10.1016/j.jmaa.2007.03.093 · Zbl 1132.35016 [6] doi:10.1006/jdeq.1999.3694 · Zbl 0959.35103 [12] doi:10.1007/PL00001368 · Zbl 0988.35095
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