Global smooth solutions for the quasilinear wave equation with boundary dissipation

We consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input-output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system. Our main tool is the geometrical analysis. The main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system. We show that the existence of global solutions depends not only on this dissipation structure but also on a Riemannian metric, given by the coefficients and the equilibrium of the system. Some geometrical conditions on this Riemannian metric are presented to guarantee the existence of global solutions. In particular, we prove that the norm of the state of the system decays exponentially if the input stops after a finite time, which implies the exponential stabilization of the system by boundary feedback.     

Asymptotic behaviour for wave equation with time-dependent coefficients

Abstract We shall find asymptotic profiles for strictly hyperbolic equations with time-dependent coefficients which are of Lipschitz class and have some stability condition. More precisely, it will be shown that there exists a solution which is not asymptotically free provided that the coefficient tends slowly to some constant. Keywords: Wave equation, Asymptotic profiles, Asymptotic integrations     

Blow up of positive initial energy solutions for a wave equation with fractional boundary dissipation

In this paper, we consider a strongly damped wave equation with fractional damping on part of its boundary and also with an internal source. Under some appropriate assumptions on the parameters, and with certain initial data, a blow-up result with positive initial energy is established.     

Asymptotic behaviour of hessian equation with boundary blow up

We consider the boundary blow up problem for k-hessian equation with nonlinearities of power and of exponential type, and prove their existence, uniqueness and asymptotic behaviour. Moreover we also show that their perturbed problem has a unique positive solution, which satisfies some asymptotic behaviors to unperturbed problems under appropriate structure hypotheses for perturbed terms.     

Decay of solutions of the wave equation with a local degenerate dissipation

We derive a precise decay estimate of the solutions to the initial-boundary value problem for the wave equation with a dissipation:u _tt ? Δu+a(x)u _t =0 in Ω × [0, ∞) with the boundary conditionu/?Ω, wherea(x) is a nonnegative function on $\bar \Omega $ satisfying $$a(x) > a.e. x \in \omega and\smallint _\omega \frac{1}{{a(x)^P }}dx< \infty for some 0< p< 1$$ for an open set $\omega \subset \bar \Omega $ including a part of ?Ω with a specific property. The result is applied to prove a global existence and decay of smooth solutions for a semilinear wave equation with such a weak dissipation.     

Asymptotic behavior of solutions for the damped wave equation with slowly decaying data

We consider the Cauchy problem for the damped wave equation

     

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.     

Asymptotic behaviour for the Kirchhoff equation

In this paper we study the asymptotic behaviour of the solution u of the Kirchhoff eqation with small data. More precisely we show that {fx293-01} every k ε Nwhere v is a suitable solution of an appropriate wave equation. Moreover we give some estimates on $$\mathop {\lim }\limits_{t \to \infty } \parallel \nabla u\parallel _2$$ .     

Asymptotic behavior for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions

We investigate a nonlinear autonomous parabolic partial differential equation in one space variable subject to Neumann boundary conditions on a compact interval. The object of our study is to determine the asymptotic behavior of solutions. Our methods are borrowed from the Liapunov theory of stability for dynamical systems. We give conditions under which a solution has a nonempty ω-limit set. We show that any such ω-limit set consists solely of equilibrium solutions. We render criteria for asymptotic stability and for instability of an equilibrium solution. We examine the possibility of escape behavior.     

Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity

Calculus of Variations and Partial Differential Equations -     

Asymptotic stability of solitary wave solutions to the regularized long-wave equation

We study the asymptotic stability of solitary wave solutions to the regularized long-wave equation (RLW) in . RLW is an equation which describes the long waves in water. To prove the result, we make use of the monotonicity of the local H¹-norm and apply the Liouville property of (RLW) as in Merle and Martel (J. Math. Pures Appl. 79 (2000) 339; Arch. Rational Mech. Anal. 157 (2001) 219).     

Periodic solutions of a quasilinear wave equation with homogeneous boundary conditions

In this paper, we prove the existence of time-periodic weak solutions for the wave equation with homogeneous boundary conditions. This paper deals with the cases where a nonlinear term has a superlinear and sublinear growth. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 189–201, 2005.     

Homogenization and behaviour of optimal controls for the wave equation in domains with oscillating boundary

In this paper the asymptotic behaviour of a second-order linear evolution problem is studied in a domain, a part of wich has an oscillating boundary. An homogeneous Neumann condition is given on the whole boundary of the domain. Moreover the behaviour of associated optimal control problem is analyzed.