The Quintic Wave Equation with Kelvin-Voigt Damping: Strichartz estimates, Well-posedness and Global Stabilization

TL;DR

This paper establishes well-posedness and exponential stabilization results for the critical quintic wave equation with Kelvin-Voigt damping in 3D, overcoming derivative loss and geometric obstructions using advanced harmonic analysis and microlocal techniques.

Contribution

It introduces a novel approach combining Littlewood-Paley analysis, Strichartz estimates, and microlocal defect measures to handle localized damping and critical nonlinearities.

Findings

Proved well-posedness for large initial data.

Established exponential energy stabilization.

Demonstrated damping effectiveness in complex geometries.

Abstract

This paper investigates the critical quintic wave equation in a 3D bounded domain subject to locally distributed Kelvin-Voigt damping. The study tackles two major mathematical challenges: the severe loss of derivatives induced by the localized thermo-viscous dissipation and the aggressive nature of the critical nonlinear term. First, we establish a robust well-posedness theory for arbitrarily large initial data. By shifting the analysis to the frequency space via a Littlewood-Paley decomposition , we employ Bernstein's inequalities to lift the damping term into an $L^2$ framework, allowing Strichartz estimates to be applied flawlessly. In the second part, we prove the uniform exponential stabilization of the energy. To overcome the reduction of the residual to the $H^{-2}$ level caused by the Kelvin-Voigt mechanism, we utilize the microlocal defect measure framework. The core of our stabilization proof relies on combining the critical Strichartz regularity $L^4_t L^{12}_x$ with a sharp Unique Continuation Property (UCP) to close the observability argument. Furthermore, we demonstrate that this microlocal mechanism is perfectly compatible with non-invasive damping geometries of arbitrarily small Lebesgue measure, successfully circumventing the geometric obstruction of trapped rays.