Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.12846

Global Well-Posedness and Energy Decay for an Abstract Viscoelastic Wave Equation with Nonlocal or Friction Damping

Jun Zhou·Teng Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper investigates an abstract viscoelastic wave equation in a separable Hilbert space, in which the damping term depends on a power of the velocity norm. Depending on whether the power exponent is zero or nonzero, this damping corresponds to friction damping or nonlocal damping, respectively; the equation also contains a viscoelastic damping represented by an integral term with a relaxation function. We establish global well-posedness for initial data in \(\mathcal{H}_1\) by combining monotone operator theory with the Faedo--Galerkin method, and prove uniqueness for all exponents greater than \(-1\), with the singular range between \(-1\) and \(0\) handled through the monotonicity rather than the Lipschitz continuity of the nonlocal damping operator. We then study the long-time behavior of the energy via integral inequalities, multiplier techniques, and Nakao's method, and obtain explicit decay estimates showing that the energy decays exponentially in the friction case and polynomially in the nonlocal case. To the best of our knowledge, this is the first work to consider the combined effects of viscoelastic damping and nonlocal damping in such an abstract framework, and it is also the first to analyze nonlocal damping with negative exponents.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jun Zhou, Teng Wang. 2026-10-09. Global Well-Posedness and Energy Decay for an Abstract Viscoelastic Wave Equation with Nonlocal or Friction Damping. https://arxiv.org/abs/2610.12846

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Global Well-Posedness of the Vacuum Free Boundary Problem for the Degenerate Compressible Navier-Stokes Equations With Large Data of Spherical Symmetry

The study of global-in-time dynamics of vacuum is crucial for understanding viscous flows. In particular, physical vacuum, which ensures a finite, nontrivial pressure-induced acceleration in the normal direction on the moving boundary, naturally arises in the motion of gaseous stars and shallow water. The corresponding large-data problems for spherically symmetric flows remain open, due to the combined difficulties of coordinate singularity at the origin and degeneracy on the moving boundary. In this paper, we analyze the free boundary problem for the barotropic compressible Navier-Stokes equations with density-dependent viscosity coefficients (as in the shallow water equations) in two and three spatial dimensions. For a general class of spherically symmetric initial densities: $ρ_0^β\in H^3$ with $β\in (\frac{1}{3},γ-1]$ ($γ$: adiabatic exponent), vanishing on the moving boundary in the form of a distance function, we establish the global existence of the unique classical solution with large data. Moreover, our solution is classical up to the moving boundary, and the physical vacuum is allowed for the data under consideration here when $β=γ-1$. The key ingredient of our analysis is to obtain the uniform control over the first-order spatial derivatives of the flow map $η(t,r)$ in spherically symmetric Lagrangian coordinates, which is achieved by some novel density-weighted estimates and flow-map-weighted estimates for the solutions.

math.AP↗
arXiv2026

Global solutions to the compressible Navier--Stokes--Poisson equations in the half-space and exterior domains

We study the initial boundary value problem for the compressible Navier--Stokes--Poisson equations without damping in the three-dimensional half-space and in smooth exterior domains. The velocity satisfies the no-slip boundary condition, and the electric potential satisfies the homogeneous Neumann boundary condition. For small perturbations of a constant state in $H^3$, with a small initial electric field in $L^2$, we establish global existence and uniqueness of strong solutions. The uniform estimates combine density dissipation, tangential and normal derivative estimates, and a modified energy expressed through the Helmholtz projection. In exterior domains, boundary normal coordinates and localized elliptic estimates control the geometric terms. We also obtain algebraic decay in the full energy norm. In the half-space, finite tangential negative Sobolev norms of order $-s$, with $\frac12\leq s<1$, yield the rate $(1+t)^{-s/2}$. In exterior domains, integrable initial density and velocity perturbations yield the same form of estimate for $0<s<\frac14$. The exterior decay proof uses whole-space negative-order estimates for localized variables and the time integrability of the dissipation.

math.AP↗
arXiv2026

Nonlocal Equations on Domains and Manifolds Without Symmetries

In this work, we establish existence and regularity results for solutions to nonlinear nonlocal equations of the form $f(Δ)u = U(x,u)$ and $f(Δ_{g})u = U(x,u)$ in Euclidean and Riemannian settings, respectively. We prove existence results across three distinct geometric scenarios: bounded domains in $\mathbb{R}^n$, perturbed equations on compact Riemannian manifolds, and noncompact Cartan-Hadamard manifolds under suitable geometric curvature conditions.

math.AP↗