NO CRITICAL NONLINEAR DIFFUSION IN 1D QUASILINEAR FULLY PARABOLIC CHEMOTAXIS SYSTEM

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 the...

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Published in:Proceedings of the American Mathematical Society 2018-06, Vol.146 (6), p.2529-2540
Main Authors: CIÉSLAK, TOMASZ, FUJIE, KENTAROU
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Language:English
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B. ANALYSIS
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description 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 the global-in-time solution. In view of our theorem one sees that the one-dimensional Keller-Segel system is essentially different from 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 classical Lyapunov functional) turns out to be rich enough to establish global existence for the initial-boundary value problem.
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title NO CRITICAL NONLINEAR DIFFUSION IN 1D QUASILINEAR FULLY PARABOLIC CHEMOTAXIS SYSTEM
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