Energy Decay Rate of Wave Equations with Indefinite Damping

Exponential Stability of Wave Equations with Potential and Indefinite Damping

First, we consider the linear wave equation utt−uxx +a(x)ut + b(x)u = 0 on a bounded interval (0, L) ⊂ R. The damping function a is allowed to change its sign. If a := 1 L R L 0 a(x)dx is positive and the spectrum of the operator (∂xx− b) is negative, exponential stability is proved for small ‖a− a‖L2 . Explicit estimates of the decay rate ω are given in terms of a and the biggest eigenvalue of...

Global Existence and Decay of Energy to Systems of Wave Equations with Damping and Supercritical Sources

This paper is concerned with a system of nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping terms. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H(Ω) × L(∂Ω...

Exponential energy decay and blow-up of solutions for a system of nonlinear viscoelastic wave equations with strong damping

with initial and Dirichlet boundary conditions. We prove that, under suitable assumptions on the functions gi, fi (i = 1, 2) and certain initial data in the stable set, the decay rate of the solution energy is exponential. Conversely, for certain initial data in the unstable set, there are solutions with positive initial energy that blow up in finite time. 2000 Mathematics Subject Classificatio...

Wave equations with non-dissipative damping

Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping

In this paper we obtain an exponential rate of decay for the solution of the viscoelastic nonlinear wave equation utt −∆u+ f(x, t, u) + ∫ t 0 g(t− τ)∆u(τ) dτ + a(x)ut = 0 in Ω× (0,∞). Here the damping term a(x)ut may be null for some part of the domain Ω. By assuming that the kernel g in the memory term decays exponentially, the damping effect allows us to avoid compactness arguments and and to...