No critical nonlinear diffusion in 1D quasilinear fully parabolic chemotaxis system

TL;DR

This paper proves that in a 1D fully parabolic chemotaxis system with a specific nonlinear diffusion, solutions exist globally regardless of initial mass, highlighting a fundamental difference from higher-dimensional cases.

Contribution

The authors demonstrate that the nonlinear diffusion 1/(1+u) is not critical in 1D, ensuring global solutions for all initial conditions, and introduce a new Lyapunov-like functional for analysis.

Findings

Solutions are globally existing for all initial conditions.

The nonlinear diffusion is not critical in 1D, unlike higher dimensions.

A new Lyapunov-like functional aids in proving global existence.

Abstract

This paper deals with the fully parabolic 1d chemotaxis system with diffusion 1/(1 + u). We prove that the above mentioned nonlinearity, despite being a natural candidate, is not critical. It means that for such a diffusion any initial condition, independently on the magnitude of mass, generates global-in-time solution. In view of our theorem one sees that one-dimensional Keller- Segel system is essentially different than its higher-dimensional versions. In order to prove our theorem we establish a new Lyapunov-like functional associated to the system. The information we gain from our new functional (together with some estimates based on the well-known old Lyapunov functional) turns out to be rich enough to establish global existence for the initial-boundary value problem.