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

Abstract

Nonlinear Klein-Gordon equation with combined power type nonlinearity and critical initial energy is investigated. The qualitative properties of a new ordinary differential equation are studied and the concavity method of Levine is improved. Necessary and sufficient conditions for finite time blow up and global existence of the solutions are proved. New sufficient conditions on the initial data for finite time blow up, based on the necessary and sufficient ones, are obtained. The asymptotic behavior of the global solutions is also investigated.

A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films

“…Let u be a solution to problem (5)- (7). Next we prove u(t) ∈ W δ for all δ ∈ (δ 1 , δ 2 ) and t ∈ [0, T ).…”

“…Proof of Theorem 2.2. Let {w j } ∞ j=1 be an orthogonal basis of H 2 0 (Ω) and an orthonormal basis of L 2 (Ω) given by eigenfunctions of ∆ 2 with the boundary condition (7). We construct the approximate solutions to problem (5)-( 7)…”

“…Therefore, we can pass to the limit in the approximate problem (10), (11). Thus u is a solution to problem ( 5)- (7) in the sense of Definition 2.1.…”

“…Proof of Theorem 2.3. As in the proof of Theorem 2.2, we construct the approximate solutions u n (t) to problem ( 5)- (7). Multiplying (10) by ξ jn (t), summing for j and integrating with respect to t, we obtain…”

“…Our main technical tool is the theory of potential wells (see e.g. [5,7,8,12,13,17,21,24,26,27,29,31]) with a slight modification, which plays an essential role in the proofs of main results.…”