On a nonlocal aggregation model with nonlinear diffusion

TL;DR

This paper studies a nonlocal aggregation model with nonlinear diffusion, proving well-posedness, smoothness, and finite-time blowup of the solution's gradient, contributing to understanding biological aggregation dynamics.

Contribution

It establishes well-posedness and smoothness results, and demonstrates finite-time gradient blowup for a nonlinear diffusion aggregation model.

Findings

Proved well-posedness and smoothness of solutions.

Identified finite-time blowup of the solution's gradient.

Analyzed the dynamics of biological aggregation models.

Abstract

We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the well-posedness, continuation criteria and smoothness of local solutions. For compactly supported nonnegative smooth initial data we prove that the gradient of the solution develops $L_x^\infty$-norm blowup in finite time.