Blowup in L ¹(Ω)-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

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.