Daniele Del Santo

Phase space analysis of partial differential equations

This collection of original articles and surveys treats the linear and nonlinear aspects of the theory of partial differential equations. Phase space analysis methods, including microlocal analysis, have yielded striking results in past years and have become one of the main tools of investigation. Equally important is their role in many applications to physics, for example, in quantum and spectral theories. Key topics: * The Cauchy problem for linear and nonlinear hyperbolic equations * Scattering theory * Inverse problems * Hyperbolic systems * Gevrey regularity of solutions of PDEs * Analytic hypoellipticity and unique features: * Original articles are self-contained with full proofs * Survey articles give a quick and direct introduction to selected topics evolving at a fast pace Graduate students at various levels as well as researchers in PDEs and related fields will find this an excellent resource.

Global existence of the solutions and formation of singularities for a class of hyperbolic systems

In this paper we prove some results concerning existence and nonexistence of global solutions of the Cauchy problem for a class of semilinear hyperbolic systems of the form

Gevrey-well-posedness for weakly hyperbolic operators with non-regular coefficients

\begin{array}{*{20}c} {\partial _t^2 u - \Delta u = Hv\left( {u,v} \right),} \\ {\partial _t^2 v - \Delta v = Hu\left( {u,v} \right),\,} \\ \end{array} {\text{in}}\,{\text{R}}^n \times \left[ {0, + \infty } \right[

Advances in Phase Space Analysis of Partial Differential Equations

(0.1) with smooth compactly supported initial data in R n. Here n ≥ 1 and H: R 2 → R is a given C 2 function. We shall call (0.1) a hyperbolic system of Hamiltonian type (see [5]). For the sake of simplicity, we shall concentrate our attention to the special case

On the Cauchy problem for hyperbolic operators with non-regular coefficients

H\left( {u,v} \right) = \frac{1} {{p + 1}}v\left| v \right|^p

The Well-Posedness Issue in Sobolev Spaces for Hyperbolic Systems with Zygmund-Type Coefficients

(0.2) with p, q > 1.