Abstract
In this paper we consider the Cauchy problem for the semilinear effectively damped wave equation
u(tt) - u(xx) + b(t)u(t) = vertical bar u vertical bar(3)mu(vertical bar u vertical bar), 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.