Uniqueness for degenerate parabolic equations in weighted $L^1$ spaces

TL;DR

This paper establishes the uniqueness of solutions to degenerate parabolic equations in bounded domains within weighted $L^1$ spaces, without boundary conditions, under specific operator assumptions and integral conditions.

Contribution

It introduces a new uniqueness result for degenerate parabolic equations in weighted $L^1$ spaces, relaxing boundary condition requirements.

Findings

Uniqueness holds for solutions satisfying an integral condition.

Applicable to possibly unbounded solutions in weighted $L^1$ spaces.

Provides conditions on the operator for uniqueness.

Abstract

We study uniqueness of solutions to degenerate parabolic problems, posed in bounded domains, where no boundary conditions are imposed. Under suitable assumptions on the operator, uniqueness is obtained for solutions that satisfy an appropriate integral condition; in particular, such condition holds for possibly unbounded solutions belonging to a suitable weighted $L^1$ space.