Scattering for the critical and localised semilinear wave equation

In this paper, we prove a scattering theorem for the critical wave equation outside convex obstacle. The proof relies on generalized Strichartz estimates.     

Critical exponent for semilinear wave equation with critical potential

We consider the Cauchy problem for the semilinear wave equation ${u_{tt} - \Delta u + V(x)u_t = |u|^p}$ .When ${V(x) = V_0(1 + |x|^2)^{-1/2}, V_0 \geq n}$ , we prove that the critical exponent for the problem is ${p_c(n)=\left\{\begin{array}{ll} 1+\frac{2}{n-1},& n \geq 2,\ +\infty,& n=1. \end{array}\right.}$      

Determination of the blow-up rate for a critical semilinear wave equation

In this paper, we determine the blow-up rate for the semilinear wave equation with critical power nonlinearity related to the conformal invariance.Mathematics Subject classification (2000): 35L05, 35L67Membre de lInstitut Universitaire de France     

The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions

In this paper, we study the global-in-time existence and the pointwise estimates of solutions to the Cauchy problem for the dissipative wave equation in multi-dimensions. Using the fixed point theorem, we obtain the global existence of the solution. In addition, the pointwise estimates of the solution are obtained by the method of the Green function. Furthermore, we obtain the Lp, 1?p?∞, convergence rate of the solution.     

Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation

The aim of this article is twofold. First we consider the wave equation in the hyperbolic space and obtain the counterparts of the Strichartz type estimates in this context. Next we examine the relationship between semilinear hyperbolic equations in the Minkowski space and in the hyperbolic space. This leads to a simple proof of the recent result of Georgiev, Lindblad and Sogge on global existence for solutions to semilinear hyperbolic problems with small data. Shifting the space-time Strichartz estimates from the hyperbolic space to the Minkowski space yields weighted Strichartz estimates in which extend the ones of Georgiev, Lindblad, and Sogge.

     

$L^p$ estimates for the linear wave equation and global existence for semilinear wave equations in exterior domains

We consider the initial-boundary value problem for the semilinear wave equation where is an exterior domain in , is a dissipative term which is effective only near the 'critical part' of the boundary. We first give some estimates for the linear equation by combining the results of the local energy decay and estimates for the Cauchy problem in the whole space. Next, on the basis of these estimates we prove global existence of small amplitude solutions for semilinear equations when is odd dimensional domain . When our result is applied if . We note that no geometrical condition on the boundary is imposed. Received April 13, 2000 / Revised July 6, 2000 / Published online February 5, 2001     

Oscillatory asymptotic expansion for semilinear wave equation

We study oscillatory properties of the solution to semilinear wave equation, assuming oscillatory terms in initial data have sufficiently small amplitude. The main result gives an a priori estimate of the remainder in the approximation by means of the method of geometric optics. The method of establishing this estimate is based on a combination between energy type estimates for transport equation and Sobolev embedding. Copyright © 2001 John Wiley & Sons, Ltd.     

L1 Decay estimates for dissipative wave equations

Let u and v be, respectively, the solutions to the Cauchy problems for the dissipative wave equation $$u_{tt}+u_t‐\Delta u=0$$\nopagenumbers\end (1) and the heat equation $$v_t‐\Delta v=0$$\nopagenumbers\end (2) We show that, as $t\rightarrow+\infty$\nopagenumbers\end , the norms $\|\partial_t^k\,D_x^\alpha u(\,\cdot\,,t)\|_{L^1({\rm R}^n)}$\nopagenumbers\end and $\|\partial_t^k\,D_x^\alpha v(\,\cdot\,,t)\|_{L^1({\rm R}^n)}$\nopagenumbers\end decay to 0 with the same polynomial rate. This result, which is well known for decay rates in $L^p({\rm R}^n)$\nopagenumbers\end with $2\leq p\leq+\infty$\nopagenumbers\end , provides another illustration of the asymptotically parabolic nature of the hyperbolic equation (1). Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

On Lp estimates for the wave equation

New and more elementary proofs are given of two results due to W. Littman: (1) Let $\text{n ? 2, p ?}\text{2n}\text{(n ? 1)}$. The estimate $\text{∫∫ (¦▽u¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{¦}{}^{\mathrm{p}}\text{) dx dt ? C ∫∫ ¦□u¦}{}^{\mathrm{p}}\text{dx dt}$ cannot hold for all u?C₀^∞(Q), Q a cube in $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">n</mi>}}\text{× R}$, some constant C. (2) Let n ? 2, p ≠ 2. The estimate $\text{∫ (¦▽(t)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(t)¦}{}^{\mathrm{p}}\text{) dx ? C(t) ∫ (¦▽u(0)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(0)¦}{}^{\mathrm{p}}\text{) dx}$ cannot hold for all C^∞ solutions of the wave equation □u = 0 in $\text{R}{}^{\mathrm{n}}\text{x R}$; all t ?$\text{R}$; some function C: $\text{R}$ → $\text{R}$.     

Characteristic space-time estimates for the wave equation

Received August 16, 1999; in final form November 25, 1999 / Published online December 8, 2000     

Decay estimates for a class of wave equations

In this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by , where is smooth away from the origin. Especially, the decay estimates for the solutions of the Klein-Gordon equation and the beam equation are simplified and slightly improved.     

Feedforward control design for a semilinear wave equation

This paper presents a flatness-based approach for the solution of the feedforward control problem for a boundary controlled semilinear wave equation. The solution to a given piecewise analytical desired output trajectory is shown to be piecewise analytical, which allows the application of a formal power series approach on suitable subregions in adequate coordinates. The subregions are determined by the characteristic curves of the system passing through the non-analyticity points of the desired trajectory. Convergence of the solution is demonstrated and simulation results are given for some set of parameters. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decay estimates for dissipative wave equations in exterior domains

Consider the initial boundary value problem for the linear dissipative wave equation (□+∂_t)u=0 in an exterior domain . Using the so-called cut-off method together with local energy decay and L² decays in the whole space, we study decay estimates of the solutions. In particular, when N?3, we derive L^p decays with p?1 of the solutions. Next, as an application of the decay estimates for the linear equation, we consider the global solvability problem for the semilinear dissipative wave equations (□+∂_t)u=f(u) with f(u)=|u|^α+1,|u|^αu in an exterior domain.     

Interior estimates for some semilinear elliptic problem with critical nonlinearity

We study compactness properties for solutions of a semilinear elliptic equation with critical nonlinearity. For high dimensions, we are able to show that any solutions sequence with uniformly bounded energy is uniformly bounded in the interior of the domain. In particular, singularly perturbed Neumann equations admit pointwise concentration phenomena only at the boundary.