Stability and blow-up result for a class of a generalized Klein-Gordon equation

Abstract

In this paper we prove the existence of solution to a generalized Klein-Gordon equation with damping and source terms. The space derivative part of the main operator is described by a pseudo-differential operator given by $-\Delta \exp (-c\Delta \cdot )$, where Δ is the Euclidean Laplace operator and c is a positive constant. To prove the existence solution we introduced an appropriate structure of Hilbert spaces which allows us to use semigroups theory when the damping term is nonlinear. Using the Nehari manifold associated to the stationary problem, we create a stable set $S$ such that, taking the initial data in $S$, the solution is global and the energy of the problem decay exponentially. In this case the damping is nonlinear and the source term satisfies the general assumption known as Ambrosetti-Rabinowitz condition. Moreover, under some appropriate conditions on the initial data we also prove a blow-up result with the source term subject to the Ambrosetti-Rabinowitz condition. Finally, we also prove a stability result with a more restrictive source term, which allows characterize the pass mountain level of the stationary problem.