# Convection-Induced Singularity Suppression in the Keller-Segel and Other Non-linear PDEs

## TL;DR

This paper demonstrates that adding a convective term with sufficiently small dissipation time can prevent blow-up in various non-linear PDEs, including the Keller-Segel model, by increasing dissipation and controlling solution growth.

## Contribution

It establishes a general link between dissipation time and blow-up prevention, providing new methods to control non-linear PDE solutions through convective terms.

## Key findings

- Blow-up can be prevented with small enough dissipation time in non-linear PDEs.
- Stationary cellular flows can have arbitrarily small dissipation times.
- Convective terms can quench reactions in ignition-type reaction-diffusion equations.

## Abstract

In this paper we study the effect of the addition of a convective term, and of the resulting increased dissipation rate, on the growth of solutions to a general class of non-linear parabolic PDEs. In particular, we show that blow-up in these models can always be prevented if the added drift has a small enough dissipation time. We also prove a general result relating the dissipation time and the effective diffusivity of stationary cellular flows, which allows us to obtain examples of simple incompressible flows with arbitrarily small dissipation times. As an application, we show that blow-up in the Keller-Segel model of chemotaxis can always be prevented if the velocity field of the ambient fluid has a sufficiently small dissipation time. We also study reaction-diffusion equations with ignition-type nonlinearities, and show that the reaction can always be quenched by the addition of a convective term with a small enough dissipation time, provided the average initial temperature is initially below the ignition threshold.