Hadamard well-posedness for a structure acoustic model with a supercritical source and damping terms

Andrew R. Becklin; Mohammad A. Rammaha

doi:10.3934/eect.2020093

Article Contents

2021, Volume 10, Issue 4: 797-836. Doi: 10.3934/eect.2020093

This issue Previous Article Uniform stability in a vectorial full Von Kármán thermoelastic system with solenoidal dissipation and free boundary conditions Next Article Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions

Hadamard well-posedness for a structure acoustic model with a supercritical source and damping terms

Andrew R. Becklin^1, , and
Mohammad A. Rammaha^2,

1.
Department of Mathematics and Computer Science, Drake University, Des Moines, IA 50311-4505, USA
2.
Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE 68588-0130, USA

^* Corresponding author: Andrew R. Becklin
^* Corresponding author: Andrew R. Becklin

Received: February 2020

Revised: June 2020

Early access: September 2020

Published: December 2021

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Abstract

This article is concerned with Hadamard's well posedness of a structural acoustic model consisting of a semilinear wave equation defined on a smooth bounded domain $ \Omega\subset\mathbb{R}^3 $ which is strongly coupled with a Berger plate equation acting only on a flat part of the boundary of $ \Omega $. The system is influenced by several competing forces. In particular, the source term acting on the wave equation is allowed to have a supercritical exponent, in the sense that its associated Nemytskii operators is not locally Lipschitz from $ H^1_{\Gamma_0}(\Omega) $ into $ L^2(\Omega) $. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions. Moreover, we prove that such solutions depend continuously on the initial data, and uniqueness is obtained in two different scenarios.

Keywords:

Structure-acoustics models,
wave-plate models,
local existence,
continuous dependence on the initial data.

Mathematics Subject Classification: Primary:35L52, 35L70;Secondary:58J45.

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