Blowup in L 1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms

Abstract

Abstract This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity u p {u}^{p} in a bounded domain Ω \Omega with the homogeneous Neumann boundary condition and positive initial values. In the case of p > 1 p\gt 1 , we prove the blowup of solutions u ( x , t ) u\left(x,t) in the sense that ‖ u ( ⋅ , t ) ‖ L 1 ( Ω ) \Vert u\left(\hspace{0.33em}\cdot \hspace{0.33em},t){\Vert }_{{L}^{1}\left(\Omega )} tends to ∞ \infty as t 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 for the blowup time. In the case of 0 < p < 1 0\lt p\lt 1 , we establish the global existence of solutions in time based on the Schauder fixed-point theorem.