Pointwise estimates and existence of solutions of porous medium and $p$-Laplace evolution equations with absorption and measure data

TL;DR

This paper establishes capacity-based necessary and sufficient conditions for the existence of solutions to certain nonlinear evolution equations, including porous medium and p-Laplace equations, with measure data and absorption terms.

Contribution

It provides new capacity criteria for existence of solutions to nonlinear PDEs with measure data, extending previous results to porous medium and p-Laplace equations.

Findings

Capacity conditions characterize solution existence.

Necessary and sufficient conditions for porous medium equations.

Sufficient conditions for p-Laplace evolution equations.

Abstract

Let $\Omega $ be a bounded domain of $\mathbb{R}^{N}(N\geq 2)$. We obtain a necessary and a sufficient condition, expressed in terms of capacities, for existence of a solution to the porous medium equation with absorption \begin{equation*} \left\{ \begin{array}{l} {u_{t}}-{\Delta }(|u|^{m-1}u)+|u|^{q-1}u=\mu ~ \text{in }\Omega \times (0,T), \\ {u}=0~~~\text{on }\partial \Omega \times (0,T), \\ u(0)=\sigma , \end{array} \right. \end{equation*} where $\sigma$ and $\mu$ are bounded Radon measures, $q>\max (m,1)$, $m>\frac{N-2}{N}$. We also obtain a sufficient condition for existence of a solution to the $p$-Laplace evolution equation \begin{equation*} \left\{ \begin{array}{l} {u_{t}}-{\Delta _{p}}u+|u|^{q-1}u=\mu ~~\text{in }\Omega \times (0,T), \\ {u}=0 ~ \text{on }\partial \Omega \times (0,T), \\ u(0)=\sigma . \end{array} \right. \end{equation*} where $q>p-1$ and $p>2$.