In this paper, we demonstrate the local well-posedness and blow up of solutions for a class of time- and space-fractional diffusion wave equation in a fractional power space associated with the Laplace operator. First, we give the definition of the solution operator which is a noteworthy extension of the solution operator of the corresponding time-fractional diffusion wave equation. We have analyzed the properties of the solution operator in the fractional power space and Lebesgue space. Next, based on some estimates of the solution operator and source term, we prove the well-posedness of mild solutions by using the contraction mapping principle. We have also investigated the blow up of solutions by using the test function method. The last result describes the properties of mild solutions when $\alpha \to 1^{-}$. The main feature of the proof is the reasonable use of continuous embedding between fractional space and Lebesgue space.

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.