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

Abstract

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 <pleq p_{1} for dgeq 3 or 1 <p for d=1, 2, where p_{1} :=1+4/(d-2). In the present paper, we give a large data blow-up of energy solution in this case, i.e. 1 <pleq p_{1} for dgeq 3 or 1 <p for d= 1, 2 (Theorem 2.4). Moreover, we also prove a non-existence of a local weak solution (Definition 2.2) in the energy-supercritical case p >p_{1} (Theorem 2.7). Our proofs are based on a invariant of a test-function method.