Catálogo de publicaciones - libros

In this review article we present an introduction to the theory of wave maps, give a short overview of some recent methods used in this field and finally we prove some recent results on ill-posedness of the corresponding Cauchy problem for the wave maps in critical Sobolev spaces. The low regularity solutions for the wave map problem are studied by the aid of appropriate bilinear estimates in the spirit of ones introduced by Klainerman and Bourgain. Our approach to obtain ill-posedness in critical Sobolev norms uses suitable family of wave maps constructed via geodesic flow on the target manifold. We give two alternative proofs of the ill-posedness: the first approach is based on the application of Fourier analysis tools, while the second proof is based on the application of the classical fundamental solution representation for the free wave equation. For the case of subcritical Sobolev norms we establish non-uniqueness of the corresponding Cauchy problem.

The aim of this work is twofold. One is to develop an approach for dealing with semilinear wave equations adopted by John [38]. In Section 2, the basis of the argument will be explained in a self-contained way. The other is an application of the approach to systems of wave equations. We shall make use of it to handle the semilinear case in Sections 3,4 and 5, and to consider the quasilinear case in Section 6. In these argument we bring such systems that the single wave components obey different propagation speeds into focus.

In this article we consider the initial-boundary value problem for linear and nonlinear wave equations in an exterior domain Ω in with the homogeneous Dirichlet boundary condition. Under the effect of localized dissipation like () we derive both of local and total energy decay estimates for the linear wave equation and apply these to the existence problem of global solutions of semilinear and quasilinear wave equations. We make no geometric condition on the shape of the boundary Ω.

The dissipation () is intended to be as weak as possible, and if the obstacle = ∖ Ω is star-shaped our results based on local energy decay hold even if () ≡ 0, while for the results concerning the total energy decay we need () ≥ > 0 near ∞.

In the final section we consider the wave equation with a ‘half-linear’ dissipation () which is like ()|| in a bounded area and which is linear like () near ∞.

We consider nonlinear wave equations with variable coefficients. Special attention is devoted to the parametric resonance phenomena.

Our aim is to describe how to obtain, with the same procedure, several results of local existence, uniqueness and propagation of regularity for the solution of a quasilinear hyperbolic Cauchy Problem.

The intention of this article is twofold: First, we survey our results from [20, 18] about energy estimates for the Cauchy problem for weakly hyperbolic operators with finite time degeneracy at time = 0. Then, in a second part, we show that these energy estimates are sharp for a wide range of examples. In particular, for these examples we precisely determine the loss of regularity that occurs in passing from the Cauchy data at = 0 to the solutions.