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September  2009, 8(5): 1541-1554. doi: 10.3934/cpaa.2009.8.1541

Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential

Zaihui Gan 1,

1. 

College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068

Received  February 2008 Revised  December 2008 Published  April 2009

In this paper, we apply a cross-constrained variational approach for the nonlinear Klein-Gordon equations with an inverse square potential in three space dimensions (which is a representative of the class of equations of interest) based on the relationship between a type of cross-constrained variational problem and energy. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we first derive a sharp threshold for global existence and blow-up of solutions to the Cauchy problem for the equations under study. On the other hand, we get an answer of the question: how small are the initial data, the global solutions exist?
Keywords: Nonlinear Klein-Gordon equation, blowup., sharp threshold, crossconstrained variational method, inverse square potential, global existence.
Mathematics Subject Classification: Primary: 35A15, 35B35; Secondary: 35Q5.
Citation: Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541
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