BLOW-UP AND GLOBAL EXISTENCE OF SOLUTIONS FOR A TIME FRACTIONAL DIFFUSION EQUATION

In this paper, we study the blow-up and global existence of solutions to the following time fractional nonlinear diffusion equations {(C)(0)D(t)(alpha)u - Delta u = I-0(t)1-gamma (vertical bar u vertical bar(p-1)u), x is an element of R-N, t > 0, u(0,x) = u(0)(x), x is an element of R-N, where 0 < alpha < gamma < 1, p > 1, u(0) is an element of C-0(R-N), I-0(t)theta denotes left Riemann-Liouville fractional integrals of order theta. (C)(0)D(t)(alpha)u = partial derivative/partial derivative t(0)I(t)(1-alpha) (u(t, x) - u(0, x)). Let beta = 1-gamma. We prove that if 1 < p < p* = max{1 + beta/alpha,1 + 2(alpha + beta)/alpha N}, the solutions of (1.1) blows up in a finite time. If N < 2(alpha + beta)/beta, p >= p * or N >= 2(alpha + beta)/beta, p > p*, and parallel to u(0)parallel to(Lqc) (R-N) is sufficiently small, where q(c) = N alpha(p-1)/2(alpha+beta), the solutions of (1.1) exists globally.