# Diffusion-driven blow-up for a non-local Fisher-KPP type model

## TL;DR

This paper investigates how diffusion-driven blow-up causes instability in a non-local Fisher-KPP model, revealing the blow-up mechanism, its rate, and resulting unstable patterns.

## Contribution

It demonstrates the occurrence of diffusion-driven blow-up and classifies its rate, providing insight into pattern formation in non-local Fisher-KPP equations.

## Key findings

- Diffusion-driven blow-up causes instability near stationary solutions.
- Blow-up rate is fully classified.
- Unstable blow-up patterns are identified and characterized.

## Abstract

The purpose of the current paper is to unveil the key mechanism which is responsible for the occurrence of {\it Turing-type instability} for a non-local Fisher-KPP type model. In particular, we prove that the solution of the considered non-local Fisher-KPP equation in the neighbourhood of a constant stationary solution, is destabilized via a {\it diffusion-driven blow-up}. It is also shown that the observed {\it diffusion-driven blow-up} is complete, whilst its blow-up rate is completely classified. Finally, the detected {\it diffusion-driven instability} results in the formation of unstable blow-up patterns, which are also identified through the determination of the blow-up profile of the solution.