Inhomogeneous Parabolic Neumann Problems

TL;DR

This paper investigates second order parabolic equations with inhomogeneous Neumann boundary conditions on Lipschitz domains, establishing existence, uniqueness, boundary regularity, and long-term behavior of solutions.

Contribution

It provides new results on the existence, uniqueness, boundary regularity, and asymptotic behavior of solutions to inhomogeneous Neumann problems for parabolic equations.

Findings

Existence and uniqueness of weak solutions

Continuity of solutions up to the boundary

Convergence to equilibrium or asymptotic almost periodicity

Abstract

We study second order parabolic equations on Lipschitz domains subject to inhomogeneous Neumann (or, more generally, Robin) boundary conditions. We prove existence and uniqueness of weak solutions and their continuity up to the boundary of the parabolic cylinder. Under natural assumptions on the coefficients and the inhomogeneity we can also prove convergence to an equilibrium or asymptotic almost periodicity.