Sharp energy criteria of blow-up for the energy-critical Klein-Gordon equation
In this paper, we study the sharp energy criteria of blow-up and global existence for the nonlinear Klein-Gordon equation by the sharp Gagliardo-Nirebergy-Sobolev inequality.
The matrix formalism previously introduced for the discussion of polymer dynamics is rendered more tractable by the introduction of an explicit operator representation satisfying boson commutation rules. Techniques are devised by which singular functions of segment coordinates may be expanded in a fluctuation series (containing ordered products of boson operators) around an equilibrium or nonequilibrium average. The techniques are here applied to equilibrium excluded‐volume expansions, and are shown to give good results. The detailed numerical results are presented in the following paper. The Gaussian potential is adopted for polymer chains in a theta solvent, but methods for its improvement are considered.