Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms

This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity $u^p$ in a bounded domain $\Om$ with the homogeneous Neumann boundary condition and positive initial values. In the case of $p>1$, we prove the blow-up of solutions $u(x,t)$ in the sense of that $\|u(\,\cdot\,,t)\|_{L^1(\Om)}$ tends to $\infty$ as $t$ approaches some value, by using a comparison principle for the corresponding ordinary differential equations and constructing special lower solutions. Moreover we provide an upper bound of the blow-up time. In the case of $0<1$, we establish the global existence of solutions in time based on the Schauder fixed point theorem.\medskip