Asymptotic behavior of solutions for linear parabolic equations with general measure data

TL;DR

This paper investigates the long-term behavior of solutions to linear parabolic equations with measure data, showing convergence to elliptic problem solutions under broad conditions.

Contribution

It establishes the asymptotic convergence of duality solutions for parabolic equations with general measure data to elliptic solutions, extending previous results to more singular measures.

Findings

Solutions converge to elliptic problem solutions as time tends to infinity.

Duality solutions exist, are unique, and exhibit asymptotic stability.

Results apply to a broad class of singular Radon measures.

Abstract

In this paper we deal with the asymptotic behavior as $t$ tends to infinity of solutions for linear parabolic equations whose model is $$ \begin{cases} u_{t}-\Delta u = \mu & \text{in}\ (0,T)\times\Omega,\\[0.7 ex] u(0,x)=u_0 & \text{in}\ \Omega, \end{cases} $$ where $\mu$ is a general, possibly singular, Radon measure which does not depend on time, and $u_0\in L^{1}(\Omega)$. We prove that the duality solution, which exists and is unique, converges to the duality solution (as introduced by G. Stampacchia) of the associated elliptic problem.