The influence of a nonlinear memory on the damped wave equation

In this paper we study the influence of a nonlinear memory F(t,u)=∫0t(t−s)−γ∣u(s,x)∣pds,γ∈(0,1), on the global existence of small data solutions to utt−△u+ut=F(t,u),u(0,x)=u0(x),ut(0,x)=u1(x), in space dimension  1≤n≤51≤n≤5. We prove the global existence for  p>p̄(n,γ), where p̄(n,γ) is the critical exponent  , i.e. no global weak solution exists for  1