Abstract
We consider the nonlinear time fractional heat equation
(c)D(0)(alpha)u(t, x) - Delta u(t,x) = vertical bar u(t, x) = vertical bar u(t, x)vertical bar(p), (t, x) is an element of (0, infinity) x D-c
subject to the initial condition
u(0, x) = u(0)(x), x is an element of D-c
and the boundary condition
u(t, x) = f(x), (t, x) is an element of (0, infinity) x partial derivative D,
where D =<(B(0, 1))over bar> is the closed unit ball in R-N, N >= 3, D-c is its complement, p > 1, u(0) is an element of L-loc(1)(<(D-c)over bar>), u(0) >= 0, f is an element of L-1(partial derivative D), integral(partial derivative D)f(x)dx > 0 and (c)D(0)(alpha)u is the Caputo fractional derivative of u of order 0 < alpha < 1 with respect to the time variable t. Using a test function approach, we obtain the critical exponent of the considered problem in the sense of Fujita. (C) 2018 Elsevier Ltd. All rights reserved.