In this paper, we investigate initial boundary value problems for Kirchhoff-type diffusion equations \begin{document}$ \partial_{t}^{\beta}u+M(\|u\|_{H_0^{s}(\Omega)}^2)(-\Delta)^{s}u = \gamma|u|^{\rho}u+g(t,x) $\end{document} with the Caputo time fractional derivatives and fractional Laplacian operators. We establish a new compactness theorem concerning time fractional derivatives. By Galerkin method, let \begin{document}$ 0<\rho<\frac{4s}{N-2s} $\end{document} when \begin{document}$ \gamma<0 $\end{document}, and \begin{document}$ 0<\rho<\min\{\frac{4s}{N},\frac{2s}{N-2s}\} $\end{document} when \begin{document}$ \gamma>0 $\end{document}, then we obtain the global existence and uniqueness of weak solutions for Kirchhoff problems. Furthermore, we get the decay properties of weak solutions in \begin{document}$ L^2(\Omega) $\end{document} and \begin{document}$ L^{\rho+2}(\Omega) $\end{document}. Remarkably, the decay rate differs from that in the case \begin{document}$ \beta = 1 $\end{document}.

In this paper, we consider initial boundary value problems for abstract fractional diffusion equations $\partial_{t}^{\beta}u+(-\Delta)^{s}u=g(t,x,u)$ with the Caputo time fractional derivatives and fractional Laplacian operators. When $g(t,x,u)$ satisfies condition (G), problems can be applied by a strong maximum principle involving time-space fractional derivatives. Hence, we establish the global existence and uniqueness of mild solution by upper and lower solutions method. Moreover, the mild solution mentioned above turns out to be a classical solution. When condition (G) does not hold, then we study nonexistence of global solutions under certain conditions, and we obtain the local existence and blow-up of mild solutions. Further, we conclude that the first eigenvalue $\lambda_1$ seems to be a critical value for nonlinear problems.

In this paper, we investigate a time-space fractional porous medium equation (FPME, for short) of the form $ \partial_{t}^{\beta}u+(-\Delta)^{s}u^m = u \displaystyle \int_\Omega u^q\,dx. $ To the best of our knowledge, there are few results concerning the global existence and blow-up of solutions for this problem. The major challenges of this paper are the presence of nonlocal terms and singularity with respect to the time-fractional derivative and space-fractional Laplacian. First, we establish the existence and $ L^\infty $ estimates of solutions for the FPME with a nonhomogeneous term. Then, applying these $ L^\infty $ estimates, the global existence and decay properties of solutions are obtained when $ q\geq q^* $ and the FPME has small initial data. Furthermore, we prove that all positive solutions blow up in finite time for $ q< q^* $. Finally, an upper bound for the lifespan of positive blow-up solutions is derived under appropriate conditions.

In this paper, we study initial boundary value problems for the following fully nonlocal Boussinesq equation $ _0^{C}D_{t}^{\beta}u+(-\Delta)^{\sigma}u+(-\Delta)^{\sigma}{_0^{C}D_{t}^{\beta}}u = -(-\Delta)^{\sigma}f(u) $ with spectral fractional Laplacian operators and Caputo fractional derivatives. To our knowledge, there are few results on fully nonlocal Boussinesq equations. The main difficulty is that each term of this equation has nonlocal effect. First, we obtain explicit expressions and some rigorous estimates of the Green operators for the corresponding linear equation. Further, we get global existence and some decay estimates of weak solutions. Second, we establish new chain and Leibnitz rules concerning $ (-\Delta)^{\sigma} $. Based on these results and small initial conditions, we obtain global existence and long-time behavior of weak solutions under different dimensions $ N $ by Banach fixed point theorem.