Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities

Abstract

In this article, we investigate the Cauchy problem for Klein-Gordon equations with combined power-type nonlinearities. Coefficients in the nonlinearities depend on the space variable. They are sign preserving functions except one of the coefficients, which may change its sign. We study completely the structure of the Nehari manifold. By using the potential well method, we give necessary and sufficient conditions for nonexistence of global solution for subcritical initial energy by means of the sign of the Nehari functional. When the energy is positive, we propose new sufficient conditions for finite time blow up of the weak solutions. One of these conditions is independent of the sign of the scalar product of the initial data. We also prove uniqueness of the weak solutions under slightly more restrictive assumptions for the powers of the nonlinearities.

On Behavior of Solutions for Nonlinear Klein–Gordon Wave Type Models with a Logarithmic Nonlinearity and Multiple Time-Varying Delays

“…The Klein-Gordon equation, with and without damping terms, has been extensively studied, either from a theoretical (as global existence and nonexistence, exponential decay of energy, time blow-up, asymptotic behavior of solutions) or from a numerical point of view, for different nonlinear source terms F(u) as F(u) = au | u | γ or logarithmic nonlinearity F(u) = au ln | u | γ , where the parameter a measures the force of the nonlinear interactions. For the first type of nonlinearities, we can cite, e.g., [2][3][4][5] and the references therein. Problems with logarithmic nonlinearity arise naturally in many areas such as quantum optics and transport phenomena, via a logarithmic Schrodinger equation (see, e.g., [6,7]); fluid dynamics via a logarithmic Korteweg-de Vries equation or a logarithmic Kadomtsev-Petviashvili equation (see, e.g., [8]); or material sciences, with a Cahn-Hilliard equation (see, e.g., [9]).…”