Global existence and blow up of the solution for nonlinear Klein–Gordon equation with general power-type nonlinearities at three initial energy levels

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

“…Now we will prove that ψ(t) > 0 for every t ∈ [b, T m ). From (19) we have that ψ(b) > 0. Otherwise from (16) it follows that γψ 2 (b) ≤ 0, which contradicts (19).…”

“…From (19) we have that ψ(b) > 0. Otherwise from (16) it follows that γψ 2 (b) ≤ 0, which contradicts (19). In order to prove that ψ(t) > 0 for every t ∈ [b, T m ) we suppose by contradiction that there exists t 0 ∈ (b, T m ) such that…”

“…Potential well method is suggested in [20] for the wave equation and further on is applied for wide class of nonlinear dispersive equations, e.g. 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).…”

“…The case of critical initial energy, i.e. E(0) = d, is treated in [5,9,26,28] for the wave and damped wave equations in bounded domains and for the Klein-Gordon equation -in [8,16,19,24]. In the above papers the global existence is proved under conditions I(0) > 0 or u(t, •) H 1 (R n ) = 0 without any restrictions on the sign of the scalar product (u 0 , u 1 ) of the initial data.…”

“…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.…”