## $$\omega$$-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

### Keywords:

rescaled initial data; rescaled solutions
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