A scale of critical exponents 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 power nonlinearity |u|p: utt−Δu+b(t)ut+m2(t)u=|u|p,t≥0,x∈Rn,u(0,x)=f(x),ut(0,x)=g(x).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 mass is dominated by the damping term, i.e. m(t)=o(b(t)) as t→∞.Then we use the obtained estimates of solutions to linear Cauchy problems to prove the existence of global in-time energy solutions to the Cauchy problem with power nonlinearity |u|p at the right-hand side of the equation, in a supercritical range p>p̄, assuming small data in the energy space (f,g)∈H1×L2. In particular, we find a threshold case for the behavior of m(t) with respect to b(t).Below the threshold, the mass has no influence on the critical exponent, so that p̄=1+4∕n, as in the case with m=0. Above the threshold, p̄=1, global (in time) small data energy solutions exist for any p>1, as it happens for the classical damped Klein–Gordon equation (b=m=1). Along the threshold, varying a parameter β∈[0,∞] which depends on the behavior of m(t) with respect to b(t), we find a scale of critical exponents, namely, p̄=1+4∕(n+4β).This scale of critical exponents is consistent with the diffusion phenomenon, that is, it is the same scale of critical exponents of the Cauchy problem for the corresponding diffusive equation −Δv+b(t)vt+m2(t)v=|v|p,t≥0,x∈Rn,v(0,x)=φ(x).