Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential (Q1041687)

The author investigates the global existence of solutions and blow-up phenomena of the three-dimensional nonlinear Klein-Gordon equations with an inverse square potential and a cubic nonlinearity: \[ \varphi_{tt}-\Delta\varphi+ \varphi+b|x|^{-2} \varphi=|\varphi|^2\varphi, \quad t\geq 0, \;x\in \mathbb R^3, \] where \(\varphi\) is an unknown complex valued function, \(b\) is a nonnegative constant, and \(\Delta\) is the Laplace operator on \(\mathbb R^3\). By constructing a type of cross-constrained variational problem and establishing cross-invariant manifolds of the evolution flow, the author derives a sharp threshold for global existence and blow-up of solutions for the equation defined on \(\mathbb R^3\).