A remark on non-existence results for the semi-linear damped Klein-Gordon equations (Harmonic Analysis and Nonlinear Partial Differential Equations)

"Harmonic Analysis and Nonlinear Partial Differential Equations". June 30～July 2, 2014. edited by Hideo Kubo and Mitsuru Sugimoto. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We consider the Cauchy problem for the semi-linear damped Klein-Gordon equations with a p-th order power nonlinearity in the Euclidean space mathbb{R}^{d}. It is well-known that the equation is locally well-posed in the energy space H^{1}(mathbb{R}^{d}) times L^{2}(mathbb{R}^{d}) in the energy-subcritical or critical case 1 p_{1} (Theorem 2.7). Our proofs are based on a invariant of a test-function method