Global nonexistence results for a class of hyperbolic systems

Our concern in this paper is to prove blow-up results to the non-autonomous nonlinear system of wave equations utt−Δu=a(t,x)|v|p,vtt−Δv=b(t,x)|u|q,t>0,x∈RN in any space dimension. We show that a curve F˜(p,q)=0 depending on the space dimension, on the exponents p,qp,q and on the behavior of the functions a(t,x)a(t,x) and b(t,x)b(t,x) exists, such that all nontrivial solutions to the above system blow-up in a finite time whenever F˜(p,q)>0. Our method of proof uses some estimates developed by Galaktionov and Pohozaev in [11] for a single non-autonomous wave equation enabling us to obtain a system of ordinary differential inequalities from which the desired result is derived. Our result generalizes some important results such as the ones in Del Santo et al. (1996) [12] and Galaktionov and Pohozaev (2003) [11]. The advantage here is that our result applies to a wide variety of problems.