Sign-preserving functionals and blow-up to Klein–Gordon equation with arbitrary high energy

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.…”

“…Remark Sufficient conditions or imply , but the contrary is not true, see Kutev et al We notice that implies , but the opposite does not occur. In the next section, we easily show this implication and by this means, we propose a condition that is better than anyone of the given in the last theorem.…”

“…To that end, we investigate the consequences of a differential inequality. First, by means of the orthogonal decomposition of the velocity, introduced in Kutev et al, where ( u , h ) 2 =0, we define the following functional: That is, This decomposition allow us to get better sufficient conditions for blow‐up. First, let us write conditions to , given in Theorem , in the following manner.…”

Blow-up of solutions for a nonlinear Petrovsky type equation with initial data at arbitrary high energy level

“…We also obtained the finite time blow-up result for certain solutions whose initial data have arbitrary high initial energy, see [20][21][22][23]. For details of the study of other kinds of evolution equations, we refer the reader to [24][25][26][27][28][29][30][31][32][33][34][35].…”