On the domain of analyticity of solutions to semilinear Klein-Gordon equations

We consider initial-value problems for semilinear Klein-Gordon equations u(tt) - Delta(x)u + u + f (u) = 0 with periodic boundary conditions. Assuming that both the initial data and the nonlinear forcing term f (u) are analytic, we provide explicit lower bounds on the decay of the radius rho(t) of analyticity of the solutions as a function of time. In particular, in one space dimension, with u real valued and f (u) = u(2k+1), we prove that the decay of rho(t) is not worse than 1/t. The results are given in a general framework, including Gevrey class solutions. (C) 2011 Elsevier Ltd. All rights reserved.