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

Abstract

In the present paper, we study the 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 \(\mathcal {D}_{0|t}^{\alpha }\) and space-fractional p-Laplacian operator \((-\varDelta )^s_p\). 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.