Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity

TL;DR

This paper investigates a one-dimensional degenerate parabolic PDE from chemotaxis modeling, establishing local well-posedness, regularity, and solution behavior, including blow-up criteria and critical mass existence.

Contribution

It introduces new analytical results for a degenerate PDE with nonlinearities, relevant to chemotaxis models, including well-posedness and solution shape insights.

Findings

Proved local in time well-posedness of the PDE.

Established regularity results and blow-up criteria.

Identified the existence of a critical mass for global solutions.

Abstract

We study a one-dimensional parabolic PDE with degenerate diffusion and non-Lipschitz nonlinearity involving the derivative. This evolution equation arises when searching radially symmetric solutions of a chemotaxis model of Patlak-Keller-Segel type. We prove its local in time wellposedness in some appropriate space, a blow-up alternative, regularity results and give an idea of the shape of solutions. A transformed and an approximate problem naturally appear in the way of the proof and are also crucial in [22] in order to study the global behaviour of solutions of the equation for a critical parameter, more precisely to show the existence of a critical mass.