Small Data Solutions for Semilinear Waves with Time-Dependent Damping and Mass Terms

Abstract

We consider the following Cauchy problem for a wave equation with time-dependent damping term b(t)ut and mass term m(t)2u, and a time-dependent non-linearity h = h(t, u): $$\displaystyle \begin {cases} u_{tt}-\Delta u+b(t)u_t+m^2(t)u=h(t,u), & t\geq 0, x\in \mathbb R^n, u(0,x)=f(x), \quad u_t(0,x)=g(x). \end {cases} $$ Here, we consider an effective time-dependent damping term and a time-dependent mass term, in the case in which the mass is dominated by the damping term, i.e. m(t) = o(b(t)) as t →∞. Under suitable assumptions on the non-linearity h = h(t, u) (Hypothesis 1.3), we prove the global existence of small data solutions in a supercritical range \(p>\bar p\), assuming small data in the energy space (f, g) ∈ H1 × L2.