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

TL;DR

This paper proves the global existence and smoothing effects for reaction-diffusion equations, specifically the porous medium equation with reaction terms on Riemannian manifolds and Euclidean space, under certain geometric and functional inequalities.

Contribution

It establishes global existence results for reaction-diffusion equations on manifolds and Euclidean space using Sobolev and Poincaré inequalities, extending previous results to more general settings.

Findings

Global solutions exist under Sobolev inequality assumptions.

Weaker conditions suffice when both Sobolev and Poincaré inequalities hold.

Results apply to porous medium equations with source terms and variable density.

Abstract

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 ${\mathbb R}^n$.