General framework to construct local-energy solutions of nonlinear diffusion equations for growing initial data

TL;DR

This paper develops a comprehensive framework for constructing local-energy solutions to nonlinear diffusion equations with growing initial data, overcoming localization challenges and applying to Finsler porous medium and fast diffusion equations.

Contribution

It introduces a systematic method to handle localization issues and nonlinear term limits in constructing solutions for general nonlinear diffusion equations.

Findings

Framework successfully constructs local-energy solutions.

Applicable to Finsler porous medium and fast diffusion equations.

Addresses localization-induced monotonicity violations.

Abstract

This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate solutions. A delicate issue for constructing local-energy solutions resides in the identification of weak limits of nonlinear terms for approximate solutions in a limiting procedure. Indeed, such an identification process often needs the maximal monotonicity of nonlinear elliptic operators (involved in the doubly-nonlinear equations) as well as uniform estimates for approximate solutions; however, even the monotonicity is violated due to a localization of the equations, which is also necessary to derive local-energy estimates for approximate solutions. In the present paper, such an inconsistency will be systematically overcome by reducing the original equation to a localized one, where a (no longer monotone) localized elliptic operator will be decomposed into the sum of a maximal monotone one and a perturbation, and by integrating all the other relevant processes. Furthermore, the general framework developed in the present paper will also be applied to the Finsler porous medium and fast diffusion equations, which are variants of the classical PME and FDE and also classified as a doubly-nonlinear equation.