Lp-asymptotic stability of 1D damped wave equations with localized and linear damping

Asymptotic Expansion for Damped Wave Equations with Periodic Coefficients

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...

Blowup and Asymptotic Stability of Weak Solutions to Wave Equations with Nonlinear Degenerate Damping and Source Terms

This article concerns the blow-up and asymptotic stability of weak solutions to the wave equation utt −∆u + |u|j(ut) = |u|p−1u in Ω× (0, T ), where p > 1 and j′ denotes the derivative of a C1 convex and real value function j. We prove that every weak solution is asymptotically stability, for every m such that 0 < m < 1, p < k + m and the the initial energy is small; the solutions blows up in fi...

Damped Wave Equations with Dynamic Boundary Conditions

We discuss several classes of linear second order initial-boundary value problems, where damping terms appear in the main wave equation as well as in the dynamic boundary condition. We investigate their wellposedness and describe some qualitative properties of their solutions, including boundedness, stability, or almost periodicity. In particular, we are able to characterize the analyticity of ...

Exponential Asymptotic Stability of Nonlinear Itô-volterra Equations with Damped Stochastic Perturbations

where K and Σ are continuous matrix–valued functions defined on R, and the functions f and g are globally linearly bounded and satisfy Lipschitz conditions. (W (t))t≥0 is a finite-dimensional standard Brownian motion. It is shown that when the entries of K are all of one sign on R, that (i) the almost sure exponential convergence of the solution to zero (ii) the p-th mean exponential convergenc...