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

In the present paper, we study the Cauchy-Dirichlet problem to the nonlocal nonlinear diffusion equation with polynomial nonlinearities $${\mathcal{D}}_{0\mathrm{\mid }t}^{\alpha }u+(-\mathrm{\Delta }{)}_p^{s}u=\gamma \mathrm{\mid }u{\mathrm{\mid }}^{m-1}u+\mu \mathrm{\mid }u{\mathrm{\mid }}^{q-2}u,\gamma ,\mu \in \mathbb{R},m>0,q>1,$$ involving time-fractional Caputo derivative ${\mathcal{D}}_{0\mathrm{\mid }t}^{\alpha }$ and space-fractional $p$-Laplacian operator $(-\mathrm{\Delta }{)}_p^s$. 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.