Semilinear Damped Wave Equation Research Articles

Global attractors for semilinear hyperbolic equations

In this paper, we use the generalized semiflow theory to study the longtime dynamical properties for a class of semilinear hyperbolic equations. The existence of global attractors is shown for the equations with no Lipschitz continuity assumption on their nonlinear terms. The results obtained here are generalizations of the related ones in Ball [Ball JM. Global attractors for damped semilinear wave equations. Discret Contin Dyn Syst 2004;10:31–52].

Global averaging and parametric resonances in damped semilinear wave equations

The long-time behaviour of solutions to a semilinear damped wave equation in a three-dimensional bounded domain with the nonlinearity rapidly oscillating in time (f = f(ε, u, t/ε)) is studied. It is proved that (under natural assumptions) the behaviour of solutions whose initial energy is not very large can be described in terms of global (uniform) attractors Aε of the corresponding dynamical processes and that, as ε → 0, these attractors tend to the global attractor A0 of the corresponding averaged system. We also give the detailed description of these attractors in the case where the limit attractor A0 is regular.Moreover, we give explicit examples of semilinear hyperbolic equations where the uniform attractor Âε (for the initial data belonging to the whole energy phase space) contains the irregular resonant part, which tends to infinity as ε → 0, and formulate the additional restrictions on the nonlinearity f which guarantee that this part is absent.

Existence and asymptotic stability for viscoelastic problems with nonlocal boundary dissipation

We consider the damped semilinear viscoelastic wave equation $$u'' - \Delta u + \int_0^t h (t - \tau ) div\{ a\nabla u(\tau )\} d\tau + g(u') = 0 in \Omega \times (0,\infty )$$ with nonlocal boundary dissipation. The existence of global solutions is proved by means of the Faedo-Galerkin method and the uniform decay rate of the energy is obtained by following the perturbed energy method provided that the kernel of the memory decays exponentially.

Convergence to equilibrium for the damped semilinear wave equation with critical exponent and dissipative boundary condition

This paper is concerned with the asymptotic behavior of the solution to the following damped semilinear wave equation with critical exponent: u t t + u t − Δ u + f ( x , u ) = 0 , ( x , t ) ∈ Ω × R + \begin{equation} u_{tt} + u_t -\Delta u + f(x,u) = 0, \qquad (x,t) \in \Omega \times \mathbb {R}^+ \end{equation} subject to the dissipative boundary condition ∂ ν u + u + u t = 0 , t > 0 , x ∈ Γ \begin{equation} \partial _\nu u+ u + u_t = 0, \qquad t > 0, \ x \in \Gamma \end{equation} and the initial conditions u | t = 0 = u 0 ( x ) , u t | t = 0 = u 1 ( x ) , x ∈ Ω , \begin{equation} u|_{t=0} = u_0(x),\quad u_t|_{t=0}=u_1(x), \qquad x \in \Omega , \end{equation} where Ω \Omega is a bounded domain in R 3 \mathbb {R}^3 with smooth boundary Γ \Gamma , ν \nu is the outward normal direction to the boundary, and f f is analytic in u u . In this paper convergence of the solution to an equilibrium as time goes to infinity is proved. While these types of results are known for the damped semilinear wave equation with interior dissipation and Dirichlet boundary condition, this is, to our knowledge, the first result with dissipative boundary condition and critical growth exponent.

Convergence to stationary solutions for a parabolic-hyperbolic phase-field system

A parabolic-hyperbolic nonconserved phase-field model is here analyzed. This isan evolution system consisting of a parabolic equation for the relative temperature$\theta$ which is nonlinearly coupled with a semilinear damped wave equation governing theorder parameter $\chi$. The latter equation is characterized by a nonlinearity $\phi(\chi)$with cubic growth. Assuming homogeneous Dirichlet and Neumann boundary conditionsfor $\theta$ and $\chi$, we prove that any weak solution has an $\omega$-limit setconsisting of one point only. This is achieved by means of adapting amethod based on the Łojasiewicz-Simon inequality.We also obtain an estimate of the decay rate to equilibrium.

A remark on the damped wave equation

In this short note we present a direct method to establish the optimal regularity of the attractor for the semilinear damped wave equation with a nonlinearity of critical growth.

THE EXPONENTIAL STABILIZATION FOR A SEMILINEAR WAVE EQUATION WITH LOCALLY DISTRIBUTED FEEDBACK

This paper considers the exponential decay of the solution to a damped semilinear wave equation with variable coefficients in the principal part by Riemannian multiplier method. A differential geometric condition that ensures the exponential decay is obtained.

Global solutions and finite time blow up for damped semilinear wave equations

A class of damped wave equations with superlinear source term is considered. It is shown that every global solution is uniformly bounded in the natural phase space. Global existence of solutions with initial data in the potential well is obtained. Finally, not only finite time blow up for solutions starting in the unstable set is proved, but also high energy initial data for which the solution blows up are constructed.

Global existence of solutions for semilinear damped wave equations in R[formula omitted] with noncompactly supported initial data

We consider the N-dimensional Cauchy problem in R N for a semilinear damped wave equation with a power-type nonlinearity | u | p . For a noncompactly supported initial data, which has a small energy, we shall derive a global in time existence result in the case when the power of the nonlinear term satisfies 1 + 2 / N < p ⩽ N / [ N - 2 ] + . This generalizes a previous result due to Todorova–Yordanov (J. Differential Equations 174 (2001) 464–489), which dealt with a solution in the framework of compactly supported initial data.

Existence and nonexistence of time-global solutions to damped wave equation on half-line

On the half-line R + = ( 0 , ∞ ) the initial-boundary value problems with null-Dirichlet boundary for both the semilinear heat equation and damped wave equation are considered. The critical exponent ρ c ( N , k ) of semilinear term for the existence and nonexistence about the semilinear heat equation on the halved space D N , k = R + k × R N - k is given by ρ c ( N , k ) = 1 + 2 / ( N + k ) (J. Appl. Math. Phys. 39 (1988) 135–149; Arch. Rational Mech. Anal. 109 (1990) 63–71). Since the damped wave equation is expected to be close to the heat equation (J. Differential Equations 191 (2003) 445–469; Math. Z. 244 (2003) 631–649), the critical exponent for the semilinear damped wave equation is expected to be same as that of the semilinear heat equation. However, there is no blow-up result on the halved space for the damped wave equation. In this paper, the exponent ρ c ( 1 , 1 ) = 2 is shown to be critical for the existence and nonexistence of time-global solution to both the semilinear heat equation and damped wave equation on the half-line R + , together with the derivation of the blow-up time. For the proof the explicit formulas of solutions are used in a similar fashion to those in Li and Zhou (Discrete Continuous Dynamic Systems 1 (1995) 503–520).

Decay estimates of solutions of a nonlinearly damped semilinear wave equation

We consider an initial boundary value problem for the equation $u_{tt}-{\mit \Delta } u-\nabla \phi \cdot \nabla u+f(u)+g(u_{t})=0$. We first prove local and global existence results under suitable conditions on $f$ and $g$. Then we show that weak solutio

Two dimensional exterior mixed problem for semilinear damped wave equations

We consider two dimensional exterior mixed problems for a semilinear damped wave equation with a power type nonlinearity | u | p . For compactly supported initial data, which have a small energy we shall derive global in time existence results in the case when the power of the nonlinearity satisfies 2 < p < + ∞ . This generalizes a previous result of [J. Differential Equations 200 (2004) 53–68], which dealt with a radially symmetric solution.

Linearization of class C1 for contractions on Banach spaces

In this work we prove a C 1 -linearization result for contraction diffeomorphisms, near a fixed point, valid in infinite-dimensional Banach spaces. As an intermediate step, we prove a specific result of existence of invariant manifolds, which can be interesting by itself and that was needed on the proof of our main theorem. Our results essentially generalize some classical results by P. Hartman in finite dimensions, and a result of Mora–Sola-Morales in the infinite-dimensional case. It is shown that the result can be applied to some abstract systems of semilinear damped wave equations.

Linearization of class C1 for contractions on Banach spaces

In this work we prove a C 1-linearization result for contraction diffeomorphisms, near a fixed point, valid in infinite-dimensional Banach spaces. As an intermediate step, we prove a specific result of existence of invariant manifolds, which can be interesting by itself and that was needed on the proof of our main theorem. Our results essentially generalize some classical results by P. Hartman in finite dimensions, and a result of Mora–Sola-Morales in the infinite-dimensional case. It is shown that the result can be applied to some abstract systems of semilinear damped wave equations.

New decay estimates for linear damped wave equations and its application to nonlinear problem

AbstractWe present new decay estimates of solutions for the mixed problem of the equation vtt−vxx+vt=0, which has the weighted initial data [v0,v1]∈(H10(0,∞) ∩L1,γ(0,∞)) × (L2(0,∞)∩L1,γ(0,∞)) (for definition of L1,γ(0,∞), see below) satisfying γ∈[0,1]. Similar decay estimates are also derived to the Cauchy problem in ℝN for utt−Δu+ut=0 with the weighted initial data. Finally, these decay estimates can be applied to the one dimensional critical exponent problem for a semilinear damped wave equation on the half line. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Global existence of solutions for semilinear damped wave equation in 2-D exterior domain

We consider a mixed problem of a damped wave equation u tt −Δ u+ u t =| u| p in the two dimensional exterior domain case. Small global in time solutions can be constructed in the case when the power p on the nonlinear term | u| p satisfies p ∗=2<p<+∞ . For this purpose we shall deal with a radially symmetric solution in the exterior domain. A new device developed in Ikehata–Matsuyama (Sci. Math. Japon. 55 (2002) 33) plays an effective role.

Asymptotic behavior of a parabolic-hyperbolic system

We consider a parabolic equation nonlinearly coupled with a damped semilinear wave equation. This system describes the evolution of the relative temperature $\vartheta$ and of the order parameter $\chi$ in a material subject to phase transitions in the framework of phase-field theories. The hyperbolic dynamics is characterized by the presence of the inertial term $\mu\partial_{t t}\chi$ with $\mu>0$. When $\mu=0$, we reduce to the well-known phase-field model of Caginalp type. The goal of the present paper is an asymptotic analysis from the viewpoint of infinite-dimensional dynamical systems. We first prove that the model, endowed with appropriate boundary conditions, generates a strongly continuous semigroup on a suitable phase-space $\mathcal V_0$, which possesses a universal attractor $\mathcal A_\mu$. Our main result establishes that $\mathcal A_\mu$ is bounded by a constant independent of $\mu$ in a smaller phase-space $\mathcal V_1$. This bound allows us to show that the lifting $\mathcal A_0$ of the universal attractor of the parabolic system (corresponding to $\mu=0$) is upper semicontinuous at $0$ with respect to the family $\{\mathcal A_\mu,\mu>0\}$. We also construct an exponential attractor; that is, a set of finite fractal dimension attracting all the trajectories exponentially fast with respect to the distance in $\mathcal V_0$. The existence of an exponential attractor is obtained in the case $\mu=0$ as well. Finally, a noteworthy consequence is that the above results also hold for the damped semilinear wave equation, which is obtained as a particular case of our system when the coupling term vanishes. This provides a generalization of a number of theorems proved in the last two decades.

Hyperbolic Equations in Uniform Spaces

The paper is devoted to the Cauchy problem for a semilinear damped wave equation in the whole of ${\mathbb R}^n$. Under suitable assumptions a bounded dissipative semigroup of global solutions is constructed in a locally uniform space $\dot H^1_{\rm lu}(

Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent

The paper is devoted to study of the longtime behavior of solutions of a damped semilinear wave equation in a bounded smooth domain of $\mathbb R^3$ with the nonautonomous external forces and with the critical cubic growth rate of the nonlinearity. In contrast to the previous papers, we prove the dissipativity of this equation in higher energy spaces $E^\alpha$, $0<\alpha\le 1$, without the usage of the dissipation integral (which is infinite in our case).

Critical exponent for semilinear damped wave equations in the N-dimensional half space

We generalize a previous result of Ikehata (Math. Methods Appl. Sci., in press), which studies the critical exponent problem of a semilinear damped wave equation in the one-dimensional half space, to the general N-dimensional half space case. That is to say, one can show the small data global existence of solutions of a mixed problem for the equation u tt − Δu+ u t =| u| p with the power p satisfying p ∗(N)=1+2/(N+1)<p⩽N/[N−2] + if we deal with the problem in the N-dimensional half space.