ON THE CRITICAL BEHAVIOR FOR TIME-FRACTIONAL REACTION DIFFUSION PROBLEMS

We first investigate the existence and nonexistence of weak solutions to the time-fractional reaction diffusion problem partial differential alpha u partial differential t alpha partial differential 2u partial differential x2 + u > x-a|u|p, t > 0, x E (0, 1], u(0, x) = u0(x), x E (0,1] under the inhomogeneous Dirichlet boundary condition u(t, 1) = (5, t > 0, where u = u(t, x), 0 < alpha < 1, partial differential alpha partial differential t alpha is the time-Caputo fractional derivative of order alpha, a > 0, p > 1 and (5 > 0. We show that, if a < 2, the existence holds for all p > 1 while if a > 2, then the dividing line with respect to existence or nonexistence is given by the critical exponent p* = a - 1. The proof of the nonexistence result is based on nonlinear capacity estimates specifically adapted to the nonlocal nature of the problem, the modified Helmholtz operator - partial differential 2 partial differential x2 + I, and the considered boundary condition. The existence part is proved by the construction of explicit solutions. We next extend our study to the case of systems.