Blow-up and global existence of solutions for time-space fractional pseudo-parabolic equation

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.