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

In this paper, we investigate the blow-up and global existence of solutions to the following time fractional nonlinear diffusion equations {(C)(0)D(t)(alpha)u - Delta u = 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 < 1, p > 1, u(0) is an element of C-0(R-N) and (C)(0)D(t)(alpha)u = (partial derivative/partial derivative t)(0)I-t(1-alpha)(u(t,x) - u(0)(x)), I-0(t)1-alpha denotes left Riemann-Liouville fractional integrals of order 1-alpha. We prove that if 1 < p < 1 + 2/N, then every nontrivial nonnegative solution blow-up in finite time, and if p >= 1 + 2/N and parallel to u(0)parallel to(Lqc(RN)), q(c) = N(p - 1)/2 is sufficiently small, then the problem has global solution.