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Nonlinear instability for a volume-filling chemotaxis model with logistic growth. (English) Zbl 1470.35361

Summary: This paper deals with a Neumann boundary value problem for a volume-filling chemotaxis model with logistic growth in a \(d\)-dimensional box \(\mathbb{T}^d = (0, \pi)^d (d = 1,2, 3)\). It is proved that given any general perturbation of magnitude \(\delta\), its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the order \(\ln(1 / \delta)\). Each initial perturbation certainly can behave drastically different from another, which gives rise to the richness of patterns.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C17 Cell movement (chemotaxis, etc.)
35K51 Initial-boundary value problems for second-order parabolic systems

References:

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