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

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