The critical exponent for a time fractional diffusion equation with nonlinear memory

In this paper, we determine the Fujita critical exponent of the following time fractional subdiffusion equation with nonlinear memory0 C D t α u − △ u =0 I t 1 − γ| u | p − 1 u , x ∈ R N , t > 0 ,u ( 0 , x ) = u 0 ( x ) , x ∈ R N ,where 0 <  α  < 1, 0 ≤  γ  < 1, γ  ≤  α , p  > 1, u 0 ∈ C 0R N,0I t 1 − γdenotes left Riemann‐Liouville fractional integral of order 1 −  γ . Let β  = 1 −  γ . We prove that, if 1 < p ≤ p ∗ = max 1 + 2 ( α + β )2 + α N − 2 ( α + β )+, 1 γ, any nontrivial positive solution blows up in a finite time. If p  >  p ∗ and ‖ u 0‖L q cR Nis sufficiently small, where q c = N α ( p − 1 ) 2 ( α + β ) , then u exists globally.