Qualitative properties of solutions to a nonlinear time-space fractional diffusion equation

In the present paper, we study the Cauchy-Dirichlet problem to a nonlocal nonlinear diffusion equation with polynomial nonlinearities D(0|t)(alpha)u + (-delta)(p)(s)u = gamma|u|(m-1)u + mu|u|(q-2)u, gamma, mu is an element of R, m > 0, q > 1, involving time-fractional Caputo derivative D alpha 0|t and space-fractional p-Laplacian operator (-delta)(p)(s). We give a simple proof of the comparison principle for the considered problem using purely algebraic relations, for different sets of gamma, mu, m and q. The Galerkin approximation method is used to prove the existence of a local weak solution. The blow-up phenomena, existence of global weak solutions and asymptotic behavior of global solutions are classified using the comparison principle.