Global Regularity versus Infinite-Time Singularity Formation in a Chemotaxis Model with Volume-Filling Effect and Degenerate Diffusion

Abstract

Abstract.A system of quasi-linear parabolic and elliptic-parabolic equations describing chemotaxis is studied. Due to the assumed presence of a volume-filling effect it is assumed that there is an impassable threshold for the density of cells. This assumption leads to singular or degenerate operators in both the diffusive and the chemotactic components of the flux of cells. We improve results from earlier works and find critical conditions which reflect the interplay between diffusion and chemotaxis and warrant that classical solutions are global in time and separated uniformly from the threshold. In the case of degenerate diffusion for the elliptic-parabolic version of the model we prove the existence of radially symmetric solutions which exhibit a phenomenon of infinite-time singularity formation in that they are global and smooth but attain the threshold in the large time limit.

A new result for global existence and boundedness of solutions to a parabolic–parabolic Keller–Segel system with logistic source

“…The following local existence result is rather standard, since a similar reasoning in [4,5,44,45,46,59]. We omit it here.…”

“…). Some recent studies show that nonlinear diffusion (Xiang [58], Viglialoro and Woolley [43], Wang et al [46], Winkler [55], Zheng [60,62]), or also (generalized) logistic dampening (Lankeit [22], Nakaguchi and Osaki [25], Viglialoro et al [40,41,42], Zheng and Wang [65]) may prevent blow-up of solutions.…”

Boundedness of solutions to a quasilinear parabolic–parabolic Keller–Segel system with a logistic source

“…The following local existence result is rather standard, since a similar reasoning in [3,7,24,28,29,30,38]. We omit it here.…”

“…Throughout this proof, C i (i ≥ 1) are positive constants depending only on some of c 0 , c 0 , c Ω , α, β, N, Ω, a, b, r, χ, u 0 and v 0 . From L 1 -boundedness of u, it suffices to prove the assertion for sufficiently large k. Now for k > 1, using L 1 -boundedness of u and following the same procedure as in ( [28], (30) and (45)), we obtain d dt…”

Analysis of a degenerate biofilm model with a nutrient taxis term

“…We would also like to point out a recent paper [20] on a chemotaxis model where singular diffusion of cells appears in the macroscopic limit of a microscopic cellular Potts model. A natural question for this class of models, whether the cell density initially below the threshold will attain this value eventually, was recently studied in [30,31].…”