Blow-up and critical exponents for nonlinear hyperbolic equations (Q1868017)

From authors' abstract: We prove nonexistence results for the Cauchy problem for an abstract hyperbolic equation in a Banach space. 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 \(u_{t}=\triangle u+b(x,t)|u|^{p}\), \(p>1,\) which are responsible for phenomena of stability, instability, blow-up and asymptotic behaviour.