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 memory {C0 D.. t u -. u = 0I1-.. t ( | u| p- 1u), x. RN, t > 0, u( 0, x) = u0( x), x. RN, where 0 <.. < 1, 0 =.. < 1,.. =.., p > 1, u0. C0 ( RN), 0I1-.. t denotes left Riemann- Liouville fractional integral of order 1 -... Let.. = 1 -... We prove that, if 1 < p = p* = max {1 + 2(.. +..) [ 2+.. N- 2(.. +..)]+, 1..}, any nontrivial positive solution blows up in a finite time. If p > p* and || u0|| Lqc ( RN) is sufficiently small, where qc = N.. ( p- 1) 2(.. +..), then u exists globally.