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\(\omega\)-limit sets for porous medium equation with initial data in some weighted spaces. (English) Zbl 1272.35041

The paper is concerned with the \(\omega\)-limit sets of the following porous medium equation \[ u_t -\Delta u^m =0, \quad \quad u(x, 0) =u_0, \] defined in \((0, \infty) \times {\mathbb{R}}^N\). Under the assumptions \(m>1\) and \(u_0 \in W_\sigma ({\mathbb{R}}^N) =\{ u: |\cdot|^\sigma u(\cdot) \in L^\infty ({\mathbb{R}}^N) \}\) with \(0<\sigma <N\), the authors establish relations between the \(\omega\)-limit sets of rescaled initial data and rescaled solutions of the equation, and an application of the results is also discussed.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
35K15 Initial value problems for second-order parabolic equations
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