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On fractional semilinear wave equations in non-cylindrical domains

Mauro Bonafini, Van Phu Cuong Le, Riccardo Molinarolo

TL;DR

This work studies the fractional semilinear wave equation $\ddot{u}+(-\Delta)^s u+\nabla W(u)=f$ in a time-dependent, non-cylindrical domain $\mathcal{O}$ with exterior Dirichlet conditions and evolving spatial domains $\{\Omega_t\}$. It develops two existence proofs under mild regularity and monotonicity assumptions: a constructive time-discretization scheme and a Lions-type penalty method, both accommodating nonlocal operators and vector-valued maps. The main results guarantee the existence of weak solutions and an energy inequality, with the energy $E(u(t))$ decreasing suitably under forcing. This extends prior cylindrical-domain results to non-cylindrical, time-evolving domains and provides robust variational methods for fractional hyperbolic problems.

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.

On fractional semilinear wave equations in non-cylindrical domains

TL;DR

This work studies the fractional semilinear wave equation in a time-dependent, non-cylindrical domain with exterior Dirichlet conditions and evolving spatial domains . It develops two existence proofs under mild regularity and monotonicity assumptions: a constructive time-discretization scheme and a Lions-type penalty method, both accommodating nonlocal operators and vector-valued maps. The main results guarantee the existence of weak solutions and an energy inequality, with the energy decreasing suitably under forcing. This extends prior cylindrical-domain results to non-cylindrical, time-evolving domains and provides robust variational methods for fractional hyperbolic problems.

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.
Paper Structure (6 sections, 2 theorems, 130 equations)

This paper contains 6 sections, 2 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Formulation of the problem and main result
  4. Time discretization method
  5. Penalty method
  6. Fractional semilinear wave equations in cylindrical domains

Key Result

Theorem 5

There exists a weak solution $u$ to problem eq:u in the sense of Definition defweaksolu. Moreover, the following energy inequality holds:

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Definition 3: Weak solution
  • Definition 4
  • Theorem 5
  • proof : Proof of Theorem \ref{['thm main result']}
  • proof : Proof of Theorem \ref{['thm main result']}
  • Definition 6
  • Definition 7
  • Theorem 8