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

Tomasz Cieślak; Kentarou Fujie

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arXiv · 1705.10203

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

Tomasz Cieślak·Kentarou Fujie

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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.

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Tomasz Cieślak, Kentarou Fujie. 2017-05-29. No critical nonlinear diffusion in 1D quasilinear fully parabolic chemotaxis system. https://arxiv.org/abs/1705.10203

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