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**Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system.**
*(English.
French summary)*
Zbl 1326.35053

The Keller-Segel chemotaxis system describes the evolution of a population of cells with density \(u\geq 0\), their motion being biased by a chemoattractant with concentration \(v\geq 0\). It reads
\[
\partial_t u = \mathrm{div} \left( \nabla u - u \nabla v \right), \qquad \tau \partial_t v = \Delta v - v + u,
\]
for \(t>0\) and \(x\in\Omega\), where \(\Omega\) is a smooth bounded domain of \(\mathbb{R}^n\), \(n\geq 2\), and \(\tau\geq 0\). It is supplemented with homogeneous Neumann boundary conditions and nonnegative initial conditions \((u_0,v_0)\in C(\overline{\Omega})\times W^{1,\infty}(\Omega)\).

An important feature of this system is that solutions blow up in finite time if \(\|u_0\|_1\) is large enough when \(n=2\) and if \(u_0\) is sufficiently concentrated when \(n\geq 3\). When \(\tau=0\), these properties were proved in [T. Nagai, Adv. Math. Sci. Appl. 5, No. 2, 581–601 (1995; Zbl 0843.92007)], but the only result available in that direction for \(\tau>0\) was the existence of self-similar solutions blowing up in finite time in [M. A. Herrero and J. J. L. Velázquez, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24, No. 4, 633–683 (1997; Zbl 0904.35037)] when \(n=2\).

The paper under review provides the first blowup result for \(\tau>0\) and \(n\geq 3\) which is valid for a large class of radially symmetric initial data. It is a milestone in the field as the technique developed therein was subsequently extended to handle the case \(n=2\) as well as variants of the Keller-Segel system. The proof relies on a careful study of the Liapunov functional available for the Keller-Segel system and more precisely on the derivation of a functional inequality relating the dissipation of the Liapunov functional to a power of it.

An important feature of this system is that solutions blow up in finite time if \(\|u_0\|_1\) is large enough when \(n=2\) and if \(u_0\) is sufficiently concentrated when \(n\geq 3\). When \(\tau=0\), these properties were proved in [T. Nagai, Adv. Math. Sci. Appl. 5, No. 2, 581–601 (1995; Zbl 0843.92007)], but the only result available in that direction for \(\tau>0\) was the existence of self-similar solutions blowing up in finite time in [M. A. Herrero and J. J. L. Velázquez, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24, No. 4, 633–683 (1997; Zbl 0904.35037)] when \(n=2\).

The paper under review provides the first blowup result for \(\tau>0\) and \(n\geq 3\) which is valid for a large class of radially symmetric initial data. It is a milestone in the field as the technique developed therein was subsequently extended to handle the case \(n=2\) as well as variants of the Keller-Segel system. The proof relies on a careful study of the Liapunov functional available for the Keller-Segel system and more precisely on the derivation of a functional inequality relating the dissipation of the Liapunov functional to a power of it.

Reviewer: Philippe Laurençot (Toulouse)

### MSC:

35B44 | Blow-up in context of PDEs |

35K51 | Initial-boundary value problems for second-order parabolic systems |

35Q92 | PDEs in connection with biology, chemistry and other natural sciences |

92C17 | Cell movement (chemotaxis, etc.) |

35K59 | Quasilinear parabolic equations |