Blow-up and critical exponents for nonlinear hyperbolic equations Original Research Article

We prove nonexistence results for the Cauchy problem for the abstract hyperbolic equation in a Banach space X, View the MathML source where View the MathML source is a C1-function. Several applications to the second- and higher-order hyperbolic equations with local and nonlocal nonlinearities are presented. We also describe an approach to Katoʹs and Johnʹs critical exponents for the semilinear equations View the MathML source, which are responsible for phenomena of stability, unstability, blow-up and asymptotic behaviour.