Global existence for reaction-diffusion evolution equations driven by the $p$-Laplacian on manifolds

TL;DR

This paper establishes conditions for the global existence and smoothing effects of reaction-diffusion equations driven by the $p$-Laplacian on noncompact manifolds like hyperbolic space, extending Euclidean methods to curved geometries.

Contribution

It provides new results on global solutions and explicit bounds for reaction-diffusion equations on noncompact manifolds with $p$-Laplacian, adapting Euclidean techniques to curved spaces.

Findings

Global existence under certain parameter and initial data conditions

Explicit $L^ ablafty$ bounds in terms of initial $L^q$ norms

Extension of Euclidean methods to hyperbolic and other noncompact manifolds

Abstract

We consider reaction-diffusion equations driven by the $p$-Laplacian on noncompact, infinite volume manifolds assumed to support the Sobolev inequality and, in some cases, to have $L^2$ spectrum bounded away from zero, the main example we have in mind being the hyperbolic space of any dimension. It is shown that, under appropriate conditions on the parameters involved and smallness conditions on the initial data, global in time solutions exist and suitable smoothing effects, namely explicit bounds on the $L^\infty$ norm of solutions at all positive times, in terms of $L^q$ norms of the data. The geometric setting discussed here requires significant modifications w.r.t. the Euclidean strategies.