Calculus of Variations and Geometric Measure Theory

G. Grillo - G. Meglioli - F. Punzo

Global existence of solutions and smoothing effects for classes of reaction-diffusion equations on manifolds

created by meglioli on 13 Oct 2022


Published Paper

Inserted: 13 oct 2022
Last Updated: 13 oct 2022

Year: 2021


We consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on $p$ and $m$ in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. Furthermore, when both the Sobolev and the Poincar\'e inequality hold, similar results hold under weaker assumptions on the forcing term. By the same functional analytic methods, we investigate global existence for solutions to the porous medium equation with source term and variable density in ${\mathbb R}^n$.