In this article, we analyze a spatio-temporally nonlocal nonlinear parabolic equation. First, we validate the equation by an existence-uniqueness result. Then, we show that blowing-up solutions exist and study their time blow-up profile. Also, a result on the existence of global solutions is presented. Furthermore, we establish necessary conditions for local or global existence.

The large time behavior of nonnegative solutions to the reaction-diffusion equation ∂ t u = −(−∆) α/2 u − u p , (α ∈ (0, 2], p > 1) posed on R N and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for p > 1 + α/N, while nonlinear effects win if p ≤ 1 + α/N.

We consider the Cauchy problem in R n , n ≥ 1, for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as t → ∞ of small data solutions have been established in the case when 1 ≤ n ≤ 3. Moreover, we derive a blow-up result under some positive data in any dimensional space.