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