Regularity of invariant sets in semilinear damped wave equations

Under fairly general assumptions, we prove that every compact invariant subset I of the semiflow generated by the semilinear damped wave equation

     

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 p|u|. 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.     

Time decay of solutions of semilinear strongly damped generalized wave equations

We study the asymptotic behaviour in time of the solutions of dissipative perturbations of wave-type equations in ?^N, u_tt + Bu_t + Au + G(u) = 0, with commuting positive operators A, B and a power like non-linearity G(u). First we give some (pseudo) conformal invariants of the linear operator in the equation. This allows us to derive optimal decay rates for the solutions of the linearized problems. We then prove some decay estimates for the non-linear problems using the tools of scattering theory and the aforementioned conformal invariants.     

Attractors for semilinear damped wave equations with an acoustic boundary condition

In this paper, we study a semilinear weakly damped wave equation equipped with an acoustic boundary condition. The problem can be considered as a system consisting of the wave equation describing the evolution of an unknown function u = u(x, t), ${x\in\Omega}$ in the domain coupled with an ordinary differential equation for an unknown function δ = δ(x, t), ${x\in\Gamma:=\partial\Omega}$ on the boundary. A compatibility condition is also added due to physical reasons. This problem is inspired on a model originally proposed by Beale and Rosencrans (Bull Am Math Soc 80:1276–1278, 1974). The goal of the paper is to analyze the global asymptotic behavior of the solutions. We prove the existence of an absorbing set and of the global attractor in the energy phase space. Furthermore, the regularity properties of the global attractor are investigated. This is a difficult issue since standard techniques based on the use of fractional operators cannot be exploited. We finally prove the existence of an exponential attractor. The analysis is carried out in dependence of two damping coefficients.     

A direct method for the stabilization of some locally damped semilinear wave equations

First, we consider a semilinear wave equation with a locally distributed damping in a bounded domain. Using the Carleman estimate, we devise an elementary proof of the exponential decay of the energy of this system. Afterwards we apply the same technique to the stabilization of the same type of equation in the whole space. Our proofs are constructive, and much simpler than those found in the literature. To cite this article: L. Tcheugoué Tébou, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On uniqueness for semilinear wave equations

 We prove that uniqueness holds in for solutions of for suitable values of s, p. This includes notably the critical case for n=4,5,6. Received: 4 February 2002 ; in final form: 9 August 2002 / Published online: 1 April 2003     

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.     

Long-time asymptotic expansion for the damped semilinear wave equation

The first initial-boundary value problem is considered for the damped semilinear wave equation with the quadratic nonlinearity. For small initial data and homogeneous boundary conditions its solution is constructed in the form of a series in the eigenfunctions of the Laplace operator. The long-time asymptotic expansion is obtained which shows the nonlinear effects of amplitude and frequency multiplication. The same results hold for the admissible initial data that are not small.     

Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity

This paper deals with the initial boundary value problem for strongly damped semilinear wave equations with logarithmic nonlinearity $u_{tt}-\Delta u-\Delta u_t={\phi }_p\left(u\right)log\left|u\right|$ in a bounded domain $\Omega \subset {\mathbb{R}}^n$. We discuss the existence, uniqueness and polynomial or exponential energy decay estimates of global weak solutions under some appropriate conditions. Moreover, we derive the finite time blow up results of weak solutions, and give the lower and upper bounds for blow-up time by the combination of the concavity method, perturbation energy method and differential–integral inequality technique.     

Simultaneous controllability of damped wave equations

We study the exact controllability of q uncoupled damped string equations by means of the same control function. This property is called simultaneous controllability. An observability inequality is proved, which implies the simultaneous controllability of the system. Our results generalize the previous results on the linear wave without the dampings. Copyright © 2016 John Wiley & Sons, Ltd.     

阻尼波动方程在模空间上的一些估计

本文研究分数次阻尼波动方程解在模空间上的估计及相应的时空估计, 作为应用, 我们将得到小初值条件下一类Cauchy问题的全局解.     

Regularity of solutions on the global attractor for a semilinear damped wave equation

We consider attractors Aη, η∈[0,1], corresponding to a singularly perturbed damped wave equation

     

On the singularities of solutions to 4-D semilinear dispersive wave equations

In this note, we are concerned with the global singularity structures of weak solutions to 4 - D semilinear dispersive wave equations whose initial data are chosen to be singular at a single point, Combining Strichartz＇s inequality with the commutator argument techniques, we show that the weak solutions stay globally conormal if the Cauchy data are conormal     

Global attractor for damped semilinear elastic beam equations with memory

Some nonlinear evolution problems arising in the theory of elastic beams with linear memory are considered. Under proper assumptions on the memory kernel the existence of uniform absorbing sets and of a global attractor is achieved.     

On self-similar solutions of semilinear wave equations in higher space dimensions

In this paper we analyze self-similar solutions of the semilinear wave equation Φ_tt − ΔΦ − Φ^p = 0 for n > 3 space dimensions. We found several classes of analytic solutions labeled by a single parameter, the form of which differ in the vicinity of the light cone. We also propose suitable numerical methods to study them.