A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up. Issue 3 (4th March 2017) - Bellomo, Nicola; Winkler, Michael
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⊂ℝ n, where χ >0 and . The main results assert the existence of a unique classical solution, extensible in time up to a maximal T max ∈(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 u r 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 | u r |, 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 m c : = ∞ 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).