Parabolic equations with exponential nonlinearity and measure data

TL;DR

This paper investigates parabolic equations with exponential nonlinearities and measure data, establishing conditions involving fractional maximal potentials for the existence of solutions in bounded domains.

Contribution

It introduces a sufficient condition based on fractional maximal potentials of measure data for solving parabolic equations with exponential nonlinearities.

Findings

Derived a criterion involving fractional maximal potentials for existence of solutions.

Extended analysis to measure data with exponential nonlinearities.

Provided conditions applicable to bounded domains in Euclidean space.

Abstract

Let $\Omega$ be a bounded domain in ${\mathbb R}^N$ and $T>0$. We study the problem \begin{equation} (P)\left\{ \begin{array}{lll} u_t - \Delta u \pm g(u) &= \mu \quad &\text{in } Q_T:=\Omega \times (0,T) \\ \phantom{------,} u&=0 &\text{on } \partial \Omega \times (0,T)\\ \phantom{----,} u(.,0) &=\omega &\text{in } \Omega. \end{array} \right. \end{equation} where $\mu$ and $\omega$ are bounded Radon measures in $Q_T$ and $\Omega$ respectively and $g(u) \sim e^{a |u|^q} $ with $a>0$ and $q \geq 1$. We provide a sufficient condition in terms of fractional maximal potentials of $\mu$ and $\omega$ for solving (P).