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

Abstract

We study the three-dimensional energy-critical quintic wave equation with localized Kelvin--Voigt damping. While the viscoelastic dissipation provides a remarkably efficient mechanism for removing energy, it simultaneously introduces a loss of derivatives that places the problem outside the classical hyperbolic framework for critical wave equations. We develop a frequency-space approach based on Littlewood--Paley theory that yields well-posedness for arbitrarily large initial data despite the derivative loss. We then establish uniform exponential stabilization by combining critical Strichartz estimates, microlocal defect measures, and the unique continuation principle of Duyckaerts, Zhang and Zuazua, obtaining observability under highly localized damping regions. The paper also identifies a structural distinction between Kelvin--Voigt damping and all previously studied lower-order dissipations. Under the critical rescaling associated with energy concentration, the Kelvin--Voigt operator becomes supercritical, revealing an obstruction absent from the classical theory. We isolate this obstruction explicitly and propose a new parabolic viewpoint suggesting that concentration inside the strictly damped region is governed by intrinsic local regularization rather than by the traditional hyperbolic concentration mechanism.