The aim of this research paper is to establish sufficient conditions for the nonexistence of global solutions for the following nonlinear fractional differential equation \begin{gather*} \mathbf{D}_{0|t}^{lpha}u+(-elta)^{\beta/2}|u|^{m-1}u+a(x)\cdotabla|u|^{q-1}u=h(x,t)|u|^p,\;\;(t,x)ı Q, u(0,x)=u_0(x),\;\;xı\R^N \end{gather*} where $(-elta)^{\beta/2}$, $0<\beta<2$ is the fractional power of $-elta$, and $\mathbf{D}_{0|t}^{\alpha}$, $(0<\alpha<1)$ denotes the time-derivative of arbitrary $\alpha\in(0;1)$ in the sense of Caputo. The results are shown by the use of test function theory and extended to systems of the same type.