Revised concavity method and application to Klein-Gordon equation

Abstract

A revised version of the concavity method of Levine, based on a new ordinary differential inequality, is proposed. Necessary and sufficient condition for nonexistence of global solutions of the inequality is proved. As an application, finite time blow up of the solution to Klein-Gordon equation with arbitrary positive initial energy is obtained under very general structural conditions.

Remarks on the qualitative behavior of the undamped Klein‐Gordon equation

“…See other works for some partial results, where the sign of I ( u 0 ) is relevant but not sufficient to characterize the behavior of the solutions. Blow‐up results for high initial energies were obtained in previous studies under sufficient conditions, very similar between them, that involve E ( u 0 , v 0 ). Here, we shall give sufficient conditions on u 0 , v 0 , to get blow‐up of solutions with positive energy, in particular for E ( u 0 , v 0 )≥ d .…”

“…This result is obtained by means of the analysis of a differential inequality. We improve the previous results in previous works . We recall some notation and results.…”

“…By means of the invariance of some sets along the solution is proved in Kutev et al that, under the assumptions of Theorem when the initial energy satisfies one of conditions to , necessarily I ( u 0 )<0. In Dimova et al, it is not proved if this implication is true or not for . Here, we do not have invariance properties, and we need to analyze when assumptions of Theorem imply I ( u 0 )<0.…”

Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy

“…for nonlinear Klein-Gordon equations see [1,18,19,24,30]. Within this method the sign of the Nehari functional I(0), see (7), is crucial for the global behavior of the solutions to (1) - (3). More precisely, for 0 < E(0) < d the solutions blow up for a finite time if I(0) < 0 and they are globally defined if I(0) ≥ 0.…”

“…E(0) > d, only partial results for global behavior of the solutions to (1) - (3) are reported in the literature. There are a few sufficient conditions on the initial data u 0 and u 1 , which guarantee finite time blow up, see [3,4,8,11,12,13,19,22,23,27,29]. In these sufficient conditions the nonnegative sign of (u 0 , u 1 ) is crucial.…”