Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping

This paper concerns with the Cauchy problem for the damped nonlinear hyperbolic system [GRAPHICS] where p, q >= 1 and satisfy pq > 1. We shall prove that if max {(1+p)/(pq-1), (1+q)/(pq-1)} < (N)/(2) for N = 1 or 3, then the solution with small initial data exists globally in time and satisfies the decay estimates parallel to u(t)parallel to(infinity) <= C(1 + t)(-sigma) and parallel to v(t)parallel to(infinity) <= C(1 + t)(-sigma)' for some positive constants a and a, while, if max {(1+p)/(pq-1), (1+q)/(pq-1)} >= (N)/(2) for N >= 1, then every solution with initial data having positive average value does not exist globally. (c) 2006 Elsevier Ltd. All rights reserved.