Weak solutions to the parabolic $p$-Laplace equation in a moving domain under a Neumann type boundary condition

TL;DR

This paper proves the existence and uniqueness of weak solutions for the parabolic p-Laplace equation in a moving domain with Neumann boundary conditions, using Galerkin methods and novel inequalities.

Contribution

It introduces a new approach to handle nonlinear gradient limits in evolving domains, extending classical results to more complex boundary conditions.

Findings

Established weak solution existence and uniqueness.

Proved strong convergence of approximate solutions.

Extended results to Leray-Lions operators.

Abstract

This paper studies the parabolic $p$-Laplace equation with $p>2$ in a moving domain under a Neumann type boundary condition corresponding to the total mass conservation. We establish the existence and uniqueness of a weak solution by the Galerkin method in evolving Bochner spaces and a monotonicity argument. The main difficulty is in characterizing the weak limit of the nonlinear gradient term, where we need to deal with a term which comes from the boundary condition and cannot be absorbed into a monotone operator. To overcome this difficulty, we prove a uniform-in-time Friedrichs type inequality on a moving domain with time-dependent basis functions and make use of it to get the strong convergence of approximate solutions. We also show that the time derivative exists in the $L^2$ sense when given data have a better regularity, and discuss extension of the existence and uniqueness results to a Leray-Lions type operator.