A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up

By Nicola Bellomo, Michael Winkler. 2016

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ABSTRACT This paper aims at providing a first step toward a qualitative theory for a new class of chemotaxis models derived from the celebrated Keller–Segel system, with the main novelty being that diffusion is nonlinear with flux delimiter features. More precisely, as a prototypical representative of this class we study radially symmetric solutions of the parabolic–elliptic system under the initial condition and no-flux boundary conditions in balls Ω⊂ℝn, where χ>0 and . The main results assert the existence of a unique classical solution, extensible in time up to a maximal Tmax∈(0,∞] which has the property that The proof of this is mainly based on comparison methods, which first relate pointwise lower and upper bounds for the spatial gradient ur to L∞ bounds for u and to upper bounds for ; second, another comparison argument involving nonlocal nonlinearities provides an appropriate control of z+ in terms of bounds for u and |ur|, with suitably mild dependence on the latter. As a consequence of (⋆), by means of suitable a priori estimates, it is moreover shown that the above solutions are global and bounded when either with if χ>1 and mc: = ∞ if χ≤1. That these conditions are essentially optimal will be shown in a forthcoming paper in which (⋆) will be used to derive complementary results on the occurrence of solutions blowing up in finite time with respect to the norm of u in L∞(Ω).

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/0f9eedaf47d67530a902c7f03b3e830a2e4c5dd4

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