Existence of weak solutions for a hyperbolic-parabolic phase field system with mixed boundary conditions on non-smooth domains

TL;DR

This paper proves the existence of weak solutions for a complex hyperbolic-parabolic system modeling damage and elasticity processes on non-smooth domains with mixed boundary conditions, using advanced variational methods.

Contribution

It introduces a novel weak formulation for hyperbolic-parabolic inclusions on Lipschitz domains and establishes existence results using time-discretization and regularization techniques.

Findings

Existence of weak solutions for the system is proven.

The approach handles non-smooth domains with mixed boundary conditions.

Variational techniques are effectively applied to recover subgradients.

Abstract

The aim of this paper is to prove existence of weak solutions of hyperbolic-parabolic evolution inclusions defined on Lipschitz domains with mixed boundary conditions describing, for instance, damage processes and elasticity with inertia terms. To this end, a suitable weak formulation to deal with such evolution inclusions in a non-smooth setting is presented. Then, existence of weak solutions is proven by utilizing time-discretization, $H^2$-regularization of the displacement variable and variational techniques from [C. Heinemann, C. Kraus: Existence results of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321--359] to recover the subgradients after the limit passages.