Semilinear Damped Klein-Gordon Models with Time-Dependent Coefficients

Abstract

We consider the following Cauchy problem for a wave equation with time-dependent damping term b(t)u _t and mass term m(t)² u, and a power nonlinearity |u|^p:

$$\displaystyle \begin {cases} u_{tt}-\varDelta u+b(t)u_t+m^2(t)u=|u|{ }^p, & t\geq 0, \ x\in \mathbb R^n,\\ u(0,x)=f(x), \quad u_t(0,x)=g(x). \end {cases} $$

We discuss how the interplay between an effective time-dependent damping term and a time-dependent mass term influences the decay rate of the solution to the corresponding linear Cauchy problem, in the case in which the damping term is dominated by the mass term, i.e. liminf_t→∞ (m(t)∕b(t)) > 1∕4.

Then we use the obtained estimates of solutions to linear Cauchy problems to prove that a unique global in-time energy solution to the Cauchy problem with power nonlinearity |u|^p at the right-hand side of the equation exists for any p > 1, assuming small data in the energy space (f, g) ∈ H ¹ × L ².