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

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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

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

Received February 2008 Revised December 2008 Published April 2009

Abstract
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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 & Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541

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