| Abstract: |
| The porous medium equation (PME) is a well studied nonlinear version of the heat equation, which reads $u_t=\Delta u^m$ for $m>1$. In particular, the diffusion becomes degenerate where the solution tends to zero, while it becomes singular where the solution tends to infinity, and compactly supported data stay with compact support at all times, even though the support eventually becomes the whole space. There is a special family of solutions, known as Barenblatts, which are explicit and, at least in Euclidean space, drive asymptotics of a quite general class of solutions. Fractional versions of the PME have also been investigated recently by J.L. Vazquez and coauthors, especially with regards to well-posedness, regularity and long-time asymptotics; one of the main issues is the fact that Barenblatts are no more explicit, and even their existence and uniqueness are nontrivial. Here I will describe some results dealing with well-posedness when finite measures are taken as initial data and asymptotics in the presence of weights. Then I will also discuss some open problems related to growing initial data and very weak solutions. |
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