Existence and non-existence of global solutions of the Cauchy problem for higher order semilinear pseudo-hyperbolic equations

We consider the Cauchy problem for a higher order pseudo-hyperbolic equation. Using the Lp→LqLp→Lq type estimation for the corresponding linear problem, the existence and non-existence criteria of global solutions are found. The existence and uniqueness of smooth global solutions are also investigated. We also establish the behavior of solutions and their derivatives as t→+∞t→+∞.