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

• Published in: Communications in partial differential equations Vol. 42; no. 3; pp. 436 - 473
• Main Authors: Bellomo, Nicola, Winkler, Michael
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 04.03.2017; Taylor & Francis Ltd
• Subjects: Blowing; Chemotaxis; degenerate diffusion; Flux; flux limitation; Mathematical models; Nonlinearity; Norms; Optimization; Partial differential equations; Primary: 35K65; Secondary: 35B45; Upper bounds
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2016.1277237

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).