Josiane Faria | Searcharxiv

arXiv2026

Asymptotic behavior of the wave equation subject to a Kelvin-Voigt nonlocal damping

In this article, we examine the well-posedness and asymptotic behavior of the energy associated with the wave equation that incorporates a Kelvin-Voigt nonlocal damping structure given by $-||\nabla u_t(t)||_2^2 \Delta u_t$. Utilizing the robust framework of nonlinear semigroups, we successfully demonstrate the existence of both strong and weak solutions. Our findings reveal that the decay rate for these solutions is optimally characterized by $1/t$, highlighting the effectiveness of this dissipative structure. This work not only enhances our understanding of the wave equation under nonlocal damping but also emphasizes the crucial balance between mathematical rigor and physical relevance.

math.AP↗

arXiv2026

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

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.

math.AP↗