A shift in the Strauss exponent for semilinear wave equations with a not effective damping

In this note we study the global existence of small data solutions to the Cauchy problem for the semilinear wave equation with a not effective scale-invariant damping term, namelyvtt-▵v+21+tvt=|v|p,v(0,x)=v0(x),vt(0,x)=v1(x), where p > 1 , n ≥ 2 We prove blow-up in finite time in the subcritical range p ∈ ( 1 , p 2 ( n ) ] and existence theorems for p > p 2 ( n ) , n = 2 , 3 In this way we find the critical exponent for small data solutions to this problem. Our results lead to the conjecture p 2 ( n ) = p 0 ( n + 2 ) for n ≥ 2 , where p 0 ( n ) is the Strauss exponent for the classical semilinear wave equation with power nonlinearity.