On a doubly nonlinear diffusion model of chemotaxis with prevention of overcrowding

Abstract

Abstract. This paper addresses the existence and regularity of weak solutions for a fully parabolic model of chemotaxis, with prevention of overcrowding, that degenerates in a two-sided fashion, including an extra nonlinearity represented by a p-Laplacian diffusion term. To prove the existence of weak solutions, a Schauder fixed-point argument is applied to a regularized problem and the compactness method is used to pass to the limit. The local Hölder regularity of weak solutions is established using the method of intrinsic scaling. The results are a contribution to showing, qualitatively, to what extent the properties of the classical Keller-Segel chemotaxis models are preserved in a more general setting. Some numerical examples illustrate the model.

Solvability of reaction–diffusion model with variable exponents

“…During the last few decades, considerable amount of work has been performed on epidemiology models [2][3][4][5][6] and their related predator-prey models (see [7][8][9] and the references therein). The cancer cell invasion disease and chemotaxis are also interesting mathematical biology models (e.g., see [10][11][12][13][14]). In relevance to the epidemic and related biological models, one can see the survey paper of Hethcote [15].…”

Adaptive Multiresolution Methods for the Simulation of Waves in Excitable Media

“…The location of the interface u = u * is not known a priori, but is part of the solution. Degenerate parabolic equations in mathematical biology were first motivated by Witelski [64] (see also [6], where a non-linear, pointwise degenerating diffusion coefficient accounts for a volume filling effect). Numerical solutions to reaction-diffusion systems with strongly degenerate diffusion exhibit a behaviour that can be surprisingly different from of the same system with a constant diffusion coefficient when parameters are chosen such that the latter case gives rise to Turing-type pattern formation [5].…”

“…In comparison with the classical FitzHugh-Nagumo kinetics (4), the Aliev-Panfilov kinetics (6) is known to provide more realistic results in the modelling of the electrical activity in ventricular tissue, for example in terms of shape of the action potential curve [52]. In both models (5) and (6), the parameters are dimensionless (as is time t ) and will be specified in Sect. 4, where numerical examples are presented.…”

“…It is the purpose of this paper to introduce two variants of a fully adaptive multiresolution (MR) scheme for the efficient computation of solutions to (1) with non-degenerate or degenerate diffusion, (2) or (3), in combination with the kinetics (5) or (6). One variant is based on a standard finite volume scheme (denoted "Scheme A") which can be applied universally to reaction-diffusion systems of the type (1); in particular, this variant can handle both kinetics (5) or (6). A second variant is based on "Scheme B", which in turn is based on Barkley's method [2] (see also [3,17]) devised for (1), (2), (5).…”

Existence and Uniqueness of Solutions of Predator-Prey Type Model with Mixed Boundary Conditions

“…Moreover the existence of weak solutions of a reaction diffusion system with mixed and no flux boundary conditions had been discussed in [5]. Bendahmane et al [7] established the weak and Holder's regularity of solutions for the doubly nonlinear chemotaxis model with Neumann boundary conditions. Weak and classical solutions to predator-prey system with cross-diffusion in heterogeneous habitats proved for Neumann boundary conditions by Bendahmane in [8].…”