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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 198, Issue 5, pp 1615–1637 | Cite as

How strong singularities can be regularized by logistic degradation in the Keller–Segel system?

  • Michael WinklerEmail author
Article
  • 129 Downloads

Abstract

The parabolic–elliptic version of the logistic Keller–Segel system given by
$$\begin{aligned} \left\{ \begin{array}{lcll} u_t &{}=&{} \Delta u - \chi \nabla \cdot (u\nabla v) + \kappa u - \mu u^2, &{}\quad x\in \varOmega , \quad t>0, \\ 0 &{}=&{} \Delta v - m(t) + u, \qquad m(t):=\frac{1}{|\varOmega |} \int _\varOmega u(x,t) \mathrm{d}x, &{}\quad x\in \varOmega , \quad t>0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$
is considered in the ball \(\varOmega =B_R(0)\subset {\mathbb {R}}^n\) with \(n\ge 1\) and \(R>0\), and with parameters \(\kappa \in {\mathbb {R}}\), \(\chi >0\) and \(\mu >0\). The focus is on the question how the zero-order dissipative term \(-\,\mu u^2\) herein, forming the apparently most essential difference between (\(\star \)) and the classical parabolic–elliptic Keller–Segel system, affects the evolution of supposedly present singular structures. For this purpose, a Neumann-type initial boundary value problem for (\(\star \)) with \(\mu >\chi \) is studied for radially decreasing nonnegative initial data \(u_0\in C^1(\overline{\varOmega }{\setminus }\{0\})\) fulfilling \(u_0(x) \le K\phi (|x|)\) for all \(x\in \overline{\varOmega }{\setminus } \{0\}\) with some \(K>0\) and some function \(\phi : (0,\infty )\rightarrow [1,\infty )\) which, besides some technical assumptions, complies with the key condition
$$\begin{aligned} \int _0^1 r^{n-1} \ln \phi (r) \mathrm{d}r <\infty . \end{aligned}$$
It is seen that for this class of data, including any such \(u_0\) satisfying
$$\begin{aligned} u_0(x) \le K e^{\lambda |x|^{-\alpha }} \qquad \text{ for } \text{ all } x\in \overline{\varOmega }{\setminus } \{0\} \end{aligned}$$
with some positive constants \(K, \lambda \) and \(\alpha <n\), the problem in question in fact admits a global solution (uv) which is smooth and classical in \(\overline{\varOmega }\times (0,\infty )\) and attains the initial data in the topology of \(C^0_{\mathrm{loc}}(\overline{\varOmega }{\setminus } \{0\})\) as \(t\searrow 0\). In view of the well-known fact that in the unperturbed Keller–Segel system already some finite-mass Radon measure-type singularities give rise to persistently singular solutions, and that hence no significant smoothing action can be expected there when infinite-mass distributions are initially present, these results reveal that zero-order quadratic degradation indeed may have a substantial effect on cross-diffusive interaction by enforcing instantaneous smoothing even of initial data exhibiting some exponentially strong singularities.

Keywords

Chemotaxis Logistic source Singularity Instantaneous smoothing 

Mathematics Subject Classification

Primary 35B65 Secondary 92C17 35Q92 35K55 35B40 

Notes

Acknowledgements

The author acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project Analysis of chemotactic cross-diffusion in complex frameworks.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PaderbornPaderbornGermany

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