Blow-up and critical exponents for nonlinear hyperbolic equations

Abstract

We prove nonexistence results for the Cauchy problem for the abstract hyperbolic equation in a Banach space X, u tt=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= Δu+b(x,t)|u| p, p>1 , which are responsible for phenomena of stability, unstability, blow-up and asymptotic behaviour.