On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback

The uniform stabilization of an originally regarded nondissipative system described by a semilinear wave equation with variable coefficients under the nonlinear boundary feedback is considered. The existence of both weak and strong solutions to the system is proven by the Galerkin method. The exponential stability of the system is obtained by introducing an equivalent energy function and using the energy multiplier method on the Riemannian manifold. This equivalent energy function shows particularly that the system is essentially a dissipative system. This result not only generalizes the result from constant coefficients to variable coefficients for these kinds of semilinear wave equations but also simplifies significantly the proof for constant coefficients case considered in [A. Guesmia, A new approach of stabilization of nondissipative distributed systems, SIAM J. Control Optim. 42 (2003) 24-52] where the system is claimed to be nondissipative.     

Solvability and stability of a semilinear wave equation with nonlinear boundary conditions

We discuss solvability for the semilinear equation of the vibrating string x_tt(t,y)−Δx(t,y)+f(t,y,x(t,y))=0 in a bounded domain, and certain type of nonlinearity on the boundary. To this effect we derive a new dual variational method. Next we discuss stability of solutions with respect to initial conditions.     

Stabilization of Euler-Bernoulli plate equation with variable coefficients by nonlinear boundary feedback

The aim of this paper is to investigate the uniform stabilization of Euler-Bernoulli plate equation with variable coefficients in the principle part subject to nonlinear boundary feedback laws. The exponential or rational energy decay rate is obtained by the multiplier method and the Riemannian geometry method.     

Stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback

We consider the stabilization of the wave equation with variable coefficients and a delay in the dissipative boundary feedback. By virtue of the Riemannian geometry methods, the energy-perturbed approach and the multiplier skills, we establish the uniform stability of the energy of the closed-loop system.     

Well-posedness and exponential stability for a nonlinear wave equation with acoustic boundary conditions

In this paper, we prove the well-posedness of a nonlinear wave equation coupled with boundary conditions of Dirichlet and acoustic type imposed on disjoints open boundary subsets. The proposed nonlinear equation models small vertical vibrations of an elastic medium with weak internal damping and a general nonlinear term. We also prove the exponential decay of the energy associated with the problem. Our results extend the ones obtained in previous results to allow weak internal dampings and removing the dimensional restriction $1\le n\le 4$. The method we use is based on a finite-dimensional approach by combining the Faedo-Galerkin method with suitable energy estimates and multiplier techniques.     

Asymptotic stability and decay rate of solutions for a nonlinear wave equation with variable coefficients

This paper is concerned with investigating the global asymptotic behavior of the solution to a nonlinear wave equation with variable coefficients. Moreover an estimate of the rate of decay of the solution is obtained.     

Blow-up of solutions to semilinear wave equations with variable coefficients and boundary

This paper is devoted to studying initial-boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions. We will establish blow-up results for the initial-boundary value problems. It is proved that there can be no global solutions no matter how small the initial data are, and also we give the life span estimate of solutions for the problems.     

On a nonlinear wave equation with boundary damping

This paper is concerned with the existence and decay of solutions of the mixed problem for the nonlinear wave equation with boundary conditions Here, Ω is an open bounded set of with boundary Γ of class C²; Γ is constituted of two disjoint closed parts Γ₀ and Γ₁ both with positive measure; the functions μ(t), f(s), g(s) satisfy the conditions μ(t) ≥ μ₀ > 0, f(s) ≥ 0, g(s) ≥ 0 for t ≥ 0, s ≥ 0 and h(x,s) is a real function where x ∈ Γ₁, ν(x) is the unit outward normal vector at x ∈ Γ₁ and α, β are non‐negative real constants. Assuming that h(x,s) is strongly monotone in s for each x ∈ Γ₁, it is proved the global existence of solutions for the previous mixed problem. For that, it is used in the Galerkin method with a special basis, the compactness approach, the Strauss approximation for real functions and the trace theorem for nonsmooth functions. The exponential decay of the energy is derived by two methods: by using a Lyapunov functional and by Nakao's method. Copyright © 2013 John Wiley & Sons, Ltd.     

On the wave equation with semilinear porous acoustic boundary conditions

The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function.     

Exponential stability of a kind of wave equation with boundary feedback control

The problem of exponential stability of a kind of wave equation with damping and boundary output feedback control is investigated. The spectral structure of the system operator is analyzed and it is shown that the c0-semigroup generated by the system operator is exponential stable if only the coefficients viscous damping and boundary feedback control are not zeros simultaneously.     

Asymptotic stability of the semilinear wave equation with boundary damping and source term

In this paper, we consider the semilinear wave equation with boundary conditions. This work is devoted to prove the uniform decay rates of the wave equation with boundary, without imposing any restrictive growth near-zero assumption on the damping term.     

General stability and exponential growth of nonlinear variable coefficient wave equation with logarithmic source and memory term

This paper is concerned with the asymptotic stability and instability of solutions to a variable coefficient logarithmic wave equation with nonlinear damping and memory term. Such model describes wave traveling through nonhomogeneous viscoelastic materials. By choosing appropriate multiplier and using weighted energy method, we prove the exponential decay of the energy. Moreover, we also obtain the instability of the solutions at the infinity in the presence of the nonlinear damping.     

A nonlinear approach to absorbing boundary conditions for the semilinear wave equation

We construct a family of absorbing boundary conditions for the semilinear wave equation. Our principal tool is the paradifferential calculus which enables us to deal with nonlinear terms. We show that the corresponding initial boundary value problems are well posed. We finally present numerical experiments illustrating the efficiency of the method.

     

Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the Kelvin–Voigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained.     

An attractor for a semilinear wave equation with boundary damping

The existence of a global compact attractor is proved. The question on the existence of a Lyapunov functional is studied. The existence of a Lyapunov functional leads to a series of important facts on the structure of the attractor. Bibliography:8 titles. Translated fromProblemy Matematicheskogo Analiza, No. 18, 1998, pp. 181–197.     

On a nonlinear wave equation with boundary conditions of two-point type

The paper is devoted to the study a nonlinear wave equation with boundary conditions of two-point type. Existence of a weak solution is proved by using Faedo-Galerkin method. Uniqueness, regularity and decay properties of solutions are also discussed.     

Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

Science China Mathematics - We consider the periodic solutions of a semilinear variable coefficient wave equation arising from the forced vibrations of a nonhomogeneous string and the propagation...