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Applicable Analysis
An International Journal
Volume 101, 2022 - Issue 9
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Research Article

On the optimality of upper estimates near blow-up in quasilinear Keller–Segel systems

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Pages 3515-3534 | Received 27 Jul 2020, Accepted 04 Nov 2020, Published online: 02 Dec 2020
 

ABSTRACT

Solutions (u,v) to the chemotaxis system ut=((u+1)m1uu(u+1)q1v),τvt=Δvv+uin a ball ΩRn, n2, wherein m,qR and τ{0,1} are given parameters with mq>−1, cannot blow up in finite time provided u is uniformly-in-time bounded in Lp(Ω) for some p>p0:=n2(1(mq)). For radially symmetric solutions, we show that, if u is only bounded in Lp0(Ω) and the technical condition m>n2p0n is fulfilled, then, for any α>np0, there is C>0 with u(x,t)C|x|αfor all xΩ and t(0,Tmax),Tmax(0,] denoting the maximal existence time. This is essentially optimal in the sense that, if this estimate held for any α<np0, then u would already be bounded in Lp(Ω) for some p>p0.

AMS Classification (2020):

Acknowledgments

The author is partially supported by the German Academic Scholarship Foundation and by the Deutsche Forschungsgemeinschaft within the project Emergence of structures and advantages in cross-diffusion systems, project number 411007140.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The author is partially supported by the Studienstiftung des Deutschen Volkes (German Academic Scholarship Foundation) and by the Deutsche Forschungsgemeinschaft within the project Emergence of structures and advantages in cross-diffusion systems, project number 411007140.

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