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

## TL;DR

This paper investigates the behavior of solutions to time-fractional diffusion equations with polynomial nonlinearities, demonstrating blowup in the $L^1$ norm for superlinear cases and global existence for sublinear cases.

## Contribution

It provides new results on blowup and global existence for time-fractional diffusion equations with polynomial terms, using comparison principles and fixed-point theorems.

## Key findings

- Solutions blow up in $L^1$ norm for $p>1$
- Global existence established for $0<p<1$
- Upper bound for blowup time derived

## 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 $\|u(\,\cdot\,,t)\|_{L^1(\Omega)}$ 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.