On fractional semilinear wave equations in non-cylindrical domains

TL;DR

This paper studies semilinear wave equations in time-evolving domains, establishing the existence of weak solutions using novel methods, and extending results to nonlocal fractional Laplacians and vector-valued maps.

Contribution

It introduces two methods for proving existence of solutions in non-cylindrical domains, including nonlocal operators and vector-valued cases, advancing the mathematical understanding of such equations.

Findings

Existence of weak solutions established under mild assumptions

Applicable to nonlocal fractional Laplacians and vector-valued maps

Two different methods demonstrated for solution construction

Abstract

In this paper, we investigate a class of semilinear wave equations in non-cylindrical time-dependent domains, subject to exterior homogeneous Dirichlet conditions. Under mild regularity and monotonicity assumptions on the evolving spatial domains, we establish existence of weak solutions by two different methods: a constructive time-discretization scheme and a penalty approach. The analysis applies to nonlocal fractional Laplacians and potentials with Lipschitz continuous gradient, and to vector-valued maps.