Smoothing effects and infinite time blowup for reaction-diffusion equations: an approach via Sobolev and Poincaré inequalities

The talk is concerned with the reaction-diffusion equation $u_t=\Delta(u^m)+u^p$, on a complete noncompact Riemannian manifold $M$. We consider the particularly delicate case when $p$ is less than $m$; moreover, we assume that the Poincaré and the Sobolev inequalities hold on $M$. We prove global existence in time of solutions for $L^m$ initial data. Furthermore, solutions are bounded for all positive times and their $L^\infty$ norm satisfy a certain quantitative bound. We also see that on a special class of Riemannian manifolds, solutions corresponding to sufficiently large $L^m$ data give rise to solutions that blow up in infinite time, a fact that cannot happen in the Euclidean setting.
The results have been recently obtained jointly with Gabriele Grillo and Fabio Punzo.