Porous medium equation with a blow-up nonlinearity and a non-decreasing constraint

TL;DR

This paper proves the existence of local strong solutions for a nonlinear porous medium equation with blow-up behavior and a non-decreasing constraint, using energy methods and variational comparison techniques.

Contribution

It introduces a novel approach to handle a fully nonlinear porous medium equation with blow-up and constraints, establishing local solution existence via approximation and variational methods.

Findings

Constructed global solutions for approximate problems using energy techniques.

Established a variational comparison principle for approximate solutions.

Proved local existence of strong solutions to the constrained nonlinear PDE.

Abstract

The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage Mechanics, is reformulated as a mixed form of two different types of doubly nonlinear evolution equations. Global (in time) solutions to some approximate problems are constructed by performing a time discretization argument and by taking advantage of energy techniques based on specific structures of the equation. Moreover, a variational comparison principle for (possibly non-unique) approximate solutions is established and it also enables us to obtain a local solution as a limit of approximate ones.