Global existence and non-existence for a higher-order parabolic equation with time-fractional term

This article is concerned with the Cauchy problem for the higher-order parabolic equation with a nonlocal in time nonlinearity: {ut+(−△)mu=1Γ(1−γ)∫0t(t−s)−γ∣u∣p(s)ds,(t,x)∈R+1×RN,u(0,x)=φ(x),x∈RN, where m,p>1m,p>1 and 0<γ<10<γ<1. By the semigroup method and the test function method, it is proved that if p>max{1+2m(2−γ)(N−2m+2mγ)+,1γ}, then the solution with small initial datum exists globally in time. If p,Np,N and mm meet some additional conditions, we can derive decay estimate ‖u(t)‖∞≤C(1+t)−σ‖u(t)‖∞≤C(1+t)−σ for some positive constant σσ. On the other hand, if p≤max{1+2m(2−γ)(N−2m+2mγ)+,1γ}, where if the equality holds, we need p>N/(N−2m)p>N/(N−2m) with N>2mN>2m. Further if the initial datum satisfies lim¯R→∞∫|x|≤Rφ(x)dx≥0, then every nontrivial weak solution does not exist globally in time.