Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions

By Tomasz Cie'slak, Christian Stinner. 2011

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Abstract In this paper we consider quasilinear Keller–Segel type systems of two kinds in higher dimensions. In the case of a nonlinear diffusion system we prove an optimal (with respect to possible nonlinear diffusions generating explosion in finite time of solutions) finite-time blowup result. In the case of a cross-diffusion system we give results which are optimal provided one assumes some proper non-decay of a nonlinear chemical sensitivity. Moreover, we show that once we do not assume the above mentioned non-decay, our result cannot be as strong as in the case of nonlinear diffusion without nonlinear cross-diffusion terms. To this end we provide an example, interesting by itself, of global-in-time unbounded solutions to the nonlinear cross-diffusion Keller–Segel system with chemical sensitivity decaying fast enough, in a range of parameters in which there is a finite-time blowup result in a corresponding case without nonlinear cross-diffusion.

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/4c506ba36435c765d42b4bcc0dcb8519943b86ae

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Citations in the corpus

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.”
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”
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.”
Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source (2015, background): “Namely, if q − p < 2 n , then all solutions are global and uniformly bounded [24, 18]; however, if q− p > 2 n , unbounded solutions do exist [22], even finite-time blow-up may occur under some additional conditions n ≥ 3 and q ≥ 1 [4, 24].”
New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models (2014, methodology): “14) only the second one requires a more detailed treatment than in [6].”
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].”
Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux (2017, background): “3) admits the blow-up solutions under some technical assumptions (see [7, 18, 20,22,27,28] and references therein).”