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

Abstract

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

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

((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

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

((0.2))

with p, q > 1.