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 Holder 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.
We study nonlinear degenerate parabolic equations where the flux function @math does not depend Lipschitz continuously on the spatial location @math . By properly adapting the "doubling of variables" device due to Kružkov [25] and Carrillo [12], we prove a uniqueness result within the class of entropy solutions for the initial value problem. We also prove a result concerning the continuous dependence on the initial data and the flux function for degenerate parabolic equations with flux function of the form @math , where @math is a vector-valued function and @math is a scalar function. The aim of this paper is to discuss the effects of linear and nonlinear diffusion in the large time asymptotic behavior of the Keller–Segel model of chemotaxis with volume filling effect. In the linear diffusion case we provide several sufficient conditions for the diffusion part to dominate and yield decay to zero solutions. We also provide an explicit decay rate towards self–similarity. Moreover, we prove that no stationary solutions with positive mass exist. In the nonlinear diffusion case we prove that the asymptotic behavior is fully determined by whether the diffusivity constant in the model is larger or smaller than the threshold value @math . Below this value we have existence of nondecaying solutions and their convergence (along subsequences) to stationary solutions. For @math all compactly supported solutions are proved to decay asymptotically to zero, unlike in the classical models with linear diffusion, where the asymptotic behavior depends on the initial mass.