Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity

Kentarou Fujie,,Institute Of Mathematics, Polish Academy Of Sciences, Warsaw, 00-656, Poland ,Tomasz Cieślak,,Department Of Mathematics, Tokyo University Of Science, Tokyo, 162-8601, Japan

doi:10.3934/dcdss.2020009

Kentarou Fujie, ,Institute Of Mathematics, Polish Academy Of Sciences, Warsaw, 00-656, Poland + Show 2 more

Open Access

https://doi.org/10.3934/dcdss.2020009

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Abstract

The paper should be viewed as complement of an earlier result in [ 10 ]. In the paper just mentioned it is shown that 1d case of a quasilinear parabolic-elliptic Keller-Segel system is very special. Namely, unlike in higher dimensions, there is no critical nonlinearity. Indeed, for the nonlinear diffusion of the form \begin{document}$ 1/u $\end{document} all the solutions, independently on the magnitude of initial mass, stay bounded. However, the argument presented in [ 10 ] deals with the Jager-Luckhaus type system. And is very sensitive to this restriction. Namely, the change of variables introduced in [ 10 ], being a main step of the method, works only for the Jager-Luckhaus modification. It does not seem to be applicable in the usual version of the parabolic-elliptic Keller-Segel system. The present paper fulfils this gap and deals with the case of the usual parabolic-elliptic version. To handle it we establish a new Lyapunov-like functional (it is related to what was done in [ 10 ]), which leads to global existence of the initial-boundary value problem for any initial mass.

Highlights

We consider the following type of PDE system ut = ∇ · (a(u)∇u − u∇v) in (0, ∞) × Ω,0 = ∆v − v + u in (0, ∞) × Ω, with a given smooth function a and Ω ⊂ Rn (n ∈ N)
The above system was introduced to describe an aggregation of cells ([14])
There are several known Lyapunov functionals associated to the above system

Summary

Introduction

The information brought by the main Lyapunov functional is often a starting point in the studies of behavior of solutions of Keller-Segel system, see for instance [1]. We can refer the reader to [18], [4] in the whole space case, and to [9] (finite-time blowup of radially symmetric solutions) and [16] (global existence for masses small enough) for bounded domains. In [10] it is shown that for any initial mass, solutions to the simplified Jager-Luckhaus system with critical diffusion remain bounded. In [8] we introduced a corresponding version of Lyapunov-like functional in the fully parabolic case.

Results

Conclusion