New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models

Abstract

We carry on our studies related to the fully parabolic quasilinear Keller-Segel system started in [6] and continued in [7]. In the above mentioned papers we proved finite-time blowup of radially symmetric solutions to the quasilinear Keller-Segel system if the nonlinear chemosensitivity is strong enough and an adequate relation between nonlinear diffusion and chemosensitivity holds. On the other hand we proved that once chemosensitivity is weak enough solutions exist globally in time. The present paper is devoted to looking for critical exponents distinguishing between those two behaviors. Moreover, we apply our results to the so-called volume filling models with a power-type probability jump function. The most important consequence of our investigations of the latter is a critical mass phenomenon found in dimension 2. Namely we find a value m * such that when the solution to the two-dimensional volume filling Keller-Segel system starts with mass smaller than m * , then it is bounded, while for initial data with mass exceeding m * solutions are unbounded, though being defined for any time t > 0.

Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation

“…For the more related works in this direction, we mention that a corresponding quasilinear version has been deeply investigated by [4,5,27,33]. Many variants of the standard Keller-Segel system (1.1) have been proposed in the past few years to model chemoatxis mechanisms in various biological contexts.…”

“…Indeed, if α < 0, the corresponding fluid-free case admits finite-time blow-up solutions (see Corollary 1.4 in[5]). …”

Large Time Behavior in a Chemotaxis Model with Nonlinear General Diffusion for Tumor Invasion

“…In [7] the authors considered the problem (1.1) with linear di¤usion term Du and showed local existence and uniqueness of nonnegative classical solutions. Recently, in this case, asymptotic behavior has been also shown in [8]; for each initial data ðu 0 ; v 0 ; w 0 ; z 0 Þ the nonnegative unique global classical solution ðu; v; w; zÞ converges to ðu 0 ; v 0 þ w 0 ; 0; c b u 0 Þ in ðL y ðWÞÞ 4 as t ! y, where f :¼ ð1=jWjÞ Ð W f .…”

“…As to this system, global existence, boundedness and blow-up are investigated in many papers (see e.g., [3,4,5,9,15,16,20]). In summary, when DðuÞ b u mÀ1 (m > 2 À 2=n, n b 1), the solution is globally bounded ( [12]); on the other hand, when DðuÞ b u mÀ1 (m < 2 À 2=n, n b 2) and W is a ball, a blow-up phenomenon occurs ( [22]).…”

Global boundedness in a quasilinear chemotaxis model with nonlinear diffusion and consumption of chemoattractant

“…Global existence, boundedness, or blowup of solutions for the systems have been studied extensively, see for example, previous studies. () Various chemotaxis models and mathematical theory of the Keller‐Segel type models have been surveyed in the studies of Bellomo et al, Hillen and Painter, Horstmann, and Winkler. ()…”