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

We consider initial-value problems for semilinear Klein–Gordon equations utt−Δxu+u+f(u)=0 with periodic boundary conditions. Assuming that both the initial data and the nonlinear forcing term f(u)f(u) are analytic, we provide explicit lower bounds on the decay of the radius ρ(t)ρ(t) of analyticity of the solutions as a function of time. In particular, in one space dimension, with uu real valued and f(u)=u2k+1f(u)=u2k+1, we prove that the decay of ρ(t)ρ(t) is not worse than 1/t1/t. The results are given in a general framework, including Gevrey class solutions.