Global existence for a Leibenson type equation with reaction on Riemannian manifolds

TL;DR

This paper proves the global existence of solutions for a nonlinear reaction-diffusion equation on certain non-compact Riemannian manifolds, establishing conditions on parameters and initial data that guarantee solutions exist for all time.

Contribution

It provides the first known global existence results for a Leibenson-type equation on non-compact manifolds with infinite volume, under specific geometric and inequality assumptions.

Findings

Global solutions exist under certain parameter conditions.

Explicit bounds on solution norms are derived.

Results differ from Euclidean cases due to manifold's non-compactness.

Abstract

We show a global existence result for a doubly nonlinear porous medium type equation of the form $$u_t = \Delta_p u^m +\, u^q$$ on a complete and non-compact Riemannian manifold $M$ of infinite volume. Here, for $1<p<N$, we assume $m(p-1)\ge1$, $m>1$ and $q>m(p-1)$. In particular, under the assumptions that $M$ supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided $q>m(p-1)+\frac pN$ and the initial datum is small enough; namely, we establish an explicit bound on the $L^\infty$ norm of the solution at all positive times, in terms of the $L^1$ norm of the data. Under the additional assumption that a Poincar\'e-type inequality also holds in $M$, we can establish the same result in the larger interval, i.e. $q>m(p-1)$. This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that $M$ is non-compact and has infinite measure.