Wang, Liangwei; Yin, Jingxue; Jin, Chunhua \(\omega\)-limit sets for porous medium equation with initial data in some weighted spaces. (English) Zbl 1272.35041 Discrete Contin. Dyn. Syst., Ser. B 18, No. 1, 223-236 (2013). 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. Reviewer: Bixiang Wang (Socorro) Cited in 7 Documents 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 PDFBibTeX XMLCite \textit{L. Wang} et al., Discrete Contin. Dyn. Syst., Ser. B 18, No. 1, 223--236 (2013; Zbl 1272.35041) Full Text: DOI