A scale of critical exponents for semilinear waves with time-dependent damping and mass terms

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 vertical bar u vertical bar(p):{u(tt) - Delta u + b(t)u(t) + m(2)(t)u = vertical bar u vertical bar(p), t >= 0, x is an element of R-n,u,(0, x) = f(x), u(t) (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 ->infinity.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 vertical bar u vertical bar(p) at the right-hand side of the equation, in a supercritical range p > (p) over bar, assuming small data in the energy space (f, g) is an element of H-1 x L-2. 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 (p) over bar = 1 + 4/n, as in the case with m = 0. Above the threshold, (p) over bar = 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 beta is an element of [0, infinity] which depends on the behavior of m(t) with respect to b(t), we find a scale of critical exponents, namely, (p) over bar = 1+ 4/(n + 4 beta).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{-Delta v + b(t)v(t) + m(2)(t)v = vertical bar u vertical bar(p), t >= 0, x is an element of R-n,v(0, x) = phi(x). (C) 2018 Elsevier Ltd. All rights reserved.