Global nonexistence results for a class of hyperbolic systems

Abstract

Our concern in this paper is to prove blow-up results to the non-autonomous nonlinear system of wave equations u t t − Δ u = a ( t , x ) | v | p , v t t − Δ v = b ( t , x ) | u | q , t > 0 , x ∈ R N in any space dimension. We show that a curve F ˜ ( p , q ) = 0 depending on the space dimension, on the exponents p , q and on the behavior of the functions a ( t , x ) and 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.