Critical regularity of nonlinearities in semilinear effectively damped wave models

In this paper we consider the Cauchy problem for the semilinear effectively damped wave equation

$$u_{tt}-u_{xx}+b(t)u_t=|u{|}^3\mu (|u|),u(0,x)=u_0(x),u_t(0,x)=u_1(x).$$

Our goal is to propose sharp conditions on $\mu$ to obtain a threshold between global (in time) existence of small data Sobolev solutions (stability of the zero solution) and blow-up behaviour even of small data Sobolev solutions.