Complicated asymptotic behavior of solutions for porous medium equation in unbounded space (Q1704544)

The authors study the complicated asymptotic behaviour for porous medium equations. It is well known that, if we have a nonnegative initial data \(u_0 \in L^1(\mathbb{R}^N)\), the solutions \(u(x,t)\) converge to the Barenblatt solution. Instead, if \(u_0 \in L^\infty(\mathbb{R}^N )\), \textit{J. L. Vázquez} and \textit{E. Zuazua} [Chin. Ann. Math., Ser. B 23, No. 2, 293--310 (2002; Zbl 1002.35020)] proved that complicated asymptotic behavior of rescaled solutions \(u(t^{\frac12} x,t)\) may occurs. The authors fully investigate the complicated asymptotic behavior of solutions for the problem \(u_t= \Delta(u^m)\) in suitable unbounded spaces. In order to reach this aim they use very sophisticated math tools such us ad-hoc propagation estimates, deep growth estimates, Moser iteration techniques and weighted \(L^1-L^\infty\) estimates.