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

Abstract

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

18 pages

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