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

By Tomasz Cie'slak, Christian Stinner. 2014

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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 some 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 jump probability 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 some initial data with mass exceeding m ⁎ solutions are unbounded, though being defined for any time t > 0 .

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/0453dae81f91319d0ec686d729773de9bc411360

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Citations in the corpus

Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux (2017, background): “On the other hand, for general m ∈ R, the existence of global bounded solutions can be ensured by m + α > 2 − 2 d (see [5,6,16,19,22,27] and references therein).”
Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities (2022, result): “These observations are quite in line with numerous findings on emergence and stabilization of singular structures ([17], [31], [34], [33], [4], [6]), and supplementary also of more subtle destabilization of spatial homogeneity ([40]), in various further among the chemotaxisproduction versions of (1.”
Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis ✩ (2016, background): “Global existence, boundedness or blowup of solutions in more general quasilinear parabolic– parabolic chemotaxis systems with nonlinear sensitivity functions and source term have been studied extensively, see for example, [11,12,25,26,34,44,48,49,59,61,62].”
Delayed blow‐up for chemotaxis models with local sensing (2020, background): “For instance, in [7, 8], the first equation is nonlinear in u and linear in v; that is, the motility and the chemosensitivity are densitydependent (but independent on the chemical concentration), which models a volume-filling effect.”
Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity (2018, background): “6) then some solutions may become unbounded ([51], [24]), where in some particular cases and related systems involving nonlinear diffusion, such unboundedness phenomena are even known to occur within finite time ([8], [9]; cf.”
Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system (2017, background): “For instance, quite a comprehensive understanding could be achieved in the case when D = D(u) > 0 and χ = χ(u) ≥ 0 are suitably smooth, independent of the signal concentration v = v(x, t) and exclusively depending on the population density u = u(x, t): Then, namely, known results indicate that essen”
How strong a logistic damping can prevent blow-up for the minimal Keller–Segel chemotaxis system? (2016, background): “1) with κ = μ = 0 admits both bounded and unbounded solutions, identified via the critical chemotactic sensitivity u 2 n [10, 20, 32].”