# The quenching of solutions to time-space fractional Kawarada problems

## TL;DR

This paper investigates the behavior of solutions to a fractional Kawarada problem, establishing conditions for quenching based on domain size and demonstrating positivity and monotonicity of solutions.

## Contribution

It introduces analysis of quenching in time-space fractional Kawarada problems, highlighting domain size effects and solution properties.

## Key findings

- Quenching solutions depend on domain size.
- Solutions remain positive and increase monotonically.
- Conditions for quenching are explicitly demonstrated.

## Abstract

Quenching solutions to a Kawarada problem with a Caputo time-fractional derivative and a fractional Laplacian are considered. The solutions to such problems may only exist locally in time when quenching occurs. Quenching and non-quenching solutions are shown to remain positive and be monotonically increasing in time under minor restrictions. Conditions for quenching to occur are demonstrated and shown to depend on the domain size.