A generalized porous medium equation related to some singular quasilinear problems

TL;DR

This paper investigates the existence and nonexistence of solutions for a generalized porous medium equation involving an infinite sum of Laplacian powers, extending classical models to include singular quasilinear problems.

Contribution

It introduces a new framework for analyzing a generalized porous medium equation with an infinite series of Laplacian terms, providing conditions for solution existence and nonexistence.

Findings

Established criteria for existence of solutions based on the sequence {a_m} and function f.

Identified conditions under which solutions do not exist for the problem.

Extended classical porous medium models to include singular quasilinear cases.

Abstract

In this paper we study existence and nonexistence of solutions for a Dirichlet boundary value problem whose model is $$ \begin{cases} -\sum_{m=1}^{\infty} a_m \Delta u^m= f&\text{in}\ \Omega \newline u=0 & \text{on}\ \partial\Omega\,, \end{cases} $$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $a_m$ is a sequence of nonnegative real numbers, and $f$ is in $L^q(\Omega)$, $q>\frac{N}{2}$.