This paper considers blow-up and global existence for a semilinear space-time fractional pseudo-parabolic equation with nonlinear memory in a bounded domain. We determine the critical exponents of the Cauchy problem when $\alpha <\gamma$ and $\alpha \ge \gamma ,$ respectively. The results obtained in this study are noteworthy extension to the results of time-fractional differential equation. The critical exponent is consistent with the corresponding Cauchy problem for the time-fractional differential equation with nonlinear memory, which illustrates that the diffusion effect of the third order term is not strong enough to change the critical exponents.

In this article, we consider the Cauchy problem for the following time-space fractional pseudo-parabolic equations

$$\{\begin{array}{l}{}_0^C{D}_t^{\alpha }(I-m\mathrm{\Delta })u+{\left(-\mathrm{\Delta }\right)}^{\frac{\beta }{2}}u=|u{|}^{p-1}u,x\in {\mathbb{R}}^N,t>0,\\ u(0,x)=u_0(x),x\in {\mathbb{R}}^N,\end{array}$$

where $0<\alpha <1,0<\beta <2,p>1,m>0,u_0\in L^q\left({\mathbb{R}}^N\right)$. An estimating $L^p-L^q$ for solution operator of time-space fractional pseudo-parabolic equations is obtained. The critical exponents of this problem are determined when $u_0\in L^q({\mathbb{R}}^N).$ Moreover, we also obtain global existence of the mild solution when $u_0\in L^p({\mathbb{R}}^N)\cap L^q({\mathbb{R}}^N)$ small enough.