Global solution for coupled nonlinear Klein-Gordon system

Abstract

The global solution for a coupled nonlinear Klein-Gordon system in twodimensional space was studied. First, a sharp threshold of blowup and global existence for the system was obtained by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow. Then the result of how small the initial data for which the solution exists globally was proved by using the scaling argument.

Blow-Up of Solutions with High Energies of a Coupled System of Hyperbolic Equations

“…Indeed, blow-up and boundedness properties hold if, ( 0 , V 0 ) < 0 and ( 0 , V 0 ) > 0, respectively; see [2]. Similar analysis have been done to prove similar characterizations for coupled systems of wave equations with linear and nonlinear damping terms; see [4][5][6][7][8] and references therein, just to cite some works of the abundant literature in the field. The qualitative analysis of the solutions with high energies is almost unknown.…”

“…Remark 5. For small energies, 0 < , the potential well method characterizes the qualitative behavior of any solution in terms of the sign of ( 0 , V 0 ); see [4][5][6][7][8]. In particular, blow-up is characterized if ( 0 , V 0 ) < 0.…”

“…Klein-Gordon systems like (KG) were studied in [2,[4][5][6][7][8][9][10][11], where blow-up results were proved. We shall illustrate how Theorem 3 is applied for each one of these systems.…”

“…5.1.7. Wu [7], Gan and Zhang [5]. By means of the potential well method, blow-up was showed for solutions of systems like (KG), with 0 < , without damping, and with linear damping, respectively.…”

“…and then, ( ( , V) , ) 2 + ( ( , V) , V) 2 − 2 ( + 1) K ( , V) = 0. (99) Hence, Theorem 3 and Corollary 4 are applied and blow-up is proved for the undamped case in [5,7], for high energies.…”