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Semilinear wave equations with time-dependent coefficients

Nenad Antonić, Matko Grbac

TL;DR

This work develops a rigorous framework for semilinear wave equations with coefficients that depend on both time and space and possess low regularity. Using Galerkin approximations combined with energy methods, the authors prove the existence of strong and weak solutions under a Lipschitz nonlinearity with a sign condition, and establish uniqueness under stronger regularity assumptions and growth constraints. The results are placed in an abstract operator-theoretic setting and then illustrated with concrete examples, including a model with symmetric and antisymmetric parts of the coefficient matrix. The methods and results have potential applications to homogenisation, microlocal energy propagation, and control problems for variable-coefficient wave equations. Overall, the paper provides a solid existence/uniqueness theory for a broad class of variable-coefficient semilinear wave problems and demonstrates how abstract results apply to practical PDE models.

Abstract

We prove the existence of strong and weak solutions to the semilinear wave equation with coefficients depending both on time and space variables, with continuous nonlinearity satisfying the sign condition. The uniqueness is proven under slightly more restrictive assumptions. Furthermore, the results obtained in abstract setting are illustrated on practical examples.

Semilinear wave equations with time-dependent coefficients

TL;DR

This work develops a rigorous framework for semilinear wave equations with coefficients that depend on both time and space and possess low regularity. Using Galerkin approximations combined with energy methods, the authors prove the existence of strong and weak solutions under a Lipschitz nonlinearity with a sign condition, and establish uniqueness under stronger regularity assumptions and growth constraints. The results are placed in an abstract operator-theoretic setting and then illustrated with concrete examples, including a model with symmetric and antisymmetric parts of the coefficient matrix. The methods and results have potential applications to homogenisation, microlocal energy propagation, and control problems for variable-coefficient wave equations. Overall, the paper provides a solid existence/uniqueness theory for a broad class of variable-coefficient semilinear wave problems and demonstrates how abstract results apply to practical PDE models.

Abstract

We prove the existence of strong and weak solutions to the semilinear wave equation with coefficients depending both on time and space variables, with continuous nonlinearity satisfying the sign condition. The uniqueness is proven under slightly more restrictive assumptions. Furthermore, the results obtained in abstract setting are illustrated on practical examples.
Paper Structure (12 sections, 6 theorems, 174 equations)

This paper contains 12 sections, 6 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Strong solutions
  4. Galërkin approximations
  5. A priori estimates
  6. The existence of a solution
  7. Weak solutions
  8. Approximating problems
  9. Energy estimates and a solution
  10. Uniqueness and some improvements in a special case
  11. Some examples and extensions
  12. Acknowledgments

Key Result

Theorem 2.2

Consider $\mathcal{R}, \mathcal{A}_0$ and $\mathcal{A}_1$ satisfying eq:ro_properties, eq:a0_properties and eq:a1_space respectively, and let ${\cal F}$ satisfy both eq:lip and eq:sign. Furthermore, take $u^0,u^1 \in V$, $f\in \textup{W}^{1,1}(0,T;H)$ and $g \in \textup{W}^{2,1}(0,T;V')$, such that with of the equation with initial conditions

Theorems & Definitions (11)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • Theorem 3.6
  • ...and 1 more