Blow-up and critical exponents for nonlinear hyperbolic equations

We prove nonexistence results for the Cauchy problem for the abstract hyperbolic equation in a Hanach space X, u(u) = f'(u), t > 0; u(0) = u(0), u(t)(0) = u(1), where f : X --> R is a C-1-function. Several applications to the second- and higher-order hyperbolic equations with local and nonlocal nonlinearities are presented. We also describe an approach to Kato's and John's critical exponents for the semilinear equations u(t) = Deltau + b(x, t)\u\(p), p > 1, which are responsible for phenomena of stability, unstability, blow-up and asymptotic behaviour. (C) 2003 Elsevier Science Ltd. All rights reserved.