Wellposedness and asymptotic behavior of solutions for the quintic wave equation with nonlocal dissipation

TL;DR

This paper studies the well-posedness and long-term behavior of solutions to a critical quintic wave equation with a novel energy-dependent damping mechanism, using advanced analytical techniques.

Contribution

It introduces a new analysis framework for a nonlinear wave model with energy-critical damping, combining Strichartz estimates and spectral methods.

Findings

Constructed weak solutions via Galerkin approximations.

Established uniform energy bounds despite critical nonlinearity.

Proved polynomial decay rates for the system's energy.

Abstract

We investigate a semilinear wave equation with energy-critical nonlinearity and a nonlinear damping mechanism driven by the total energy of the system. The model combines the quintic defocusing term with a time-dependent dissipation of the form E(t)u_t, which introduces a nonstandard feedback structure coupling the dynamics and the energy functional. Weak solutions are constructed via Galerkin approximations, with the passage to the limit relying on uniform energy estimates and compactness arguments. Special attention is devoted to the critical nature of the nonlinearity, where concentration phenomena prevent purely energy-based methods from yielding refined spacetime control. This difficulty is resolved by incorporating nonhomogeneous Strichartz estimates together with smoothly truncated spectral approximations, ensuring uniform bound at the dispersive level. Finally, we establish polynomial decay rates for the energy by adapting Nakao's method to the present nonlinear dissipative framework. The results highlight the stabilizing effect of the energy-dependent damping and its interaction with the critical wave dynamics.