Non autonomous parabolic problems with unbounded coefficients in unbounded domains

TL;DR

This paper establishes existence, uniqueness, and qualitative properties of evolution operators for nonautonomous elliptic operators with unbounded coefficients in unbounded domains, under various boundary conditions.

Contribution

It provides new results on the well-posedness and qualitative analysis of nonautonomous elliptic problems with unbounded coefficients in unbounded domains.

Findings

Existence and uniqueness of the evolution operator in bounded and continuous functions.

Qualitative properties and compactness of the evolution operator.

Uniform gradient estimates for solutions.

Abstract

Given a class of nonautonomous elliptic operators $\A(t)$ with unbounded coefficients, defined in $\overline{I \times \Om}$ (where $I$ is a right-halfline or $I=\R$ and $\Om\subset \Rd$ is possibly unbounded), we prove existence and uniqueness of the evolution operator associated to $\A(t)$ in the space of bounded and continuous functions, under Dirichlet and first order, non tangential homogeneous boundary conditions. Some qualitative properties of the solutions, the compactness of the evolution operator and some uniform gradient estimates are then proved.