Refined Description and Stability for Singular Solutions of the 2D Keller‐Segel System

Abstract

We construct solutions to the two‐dimensional parabolic‐elliptic Keller‐Segel model for chemotaxis that blow up in finite time T. The solution is decomposed as the sum of a stationary state concentrated at scale λ and of a perturbation. We rely on a detailed spectral analysis for the linearised dynamics in the parabolic neighbourhood of the singularity performed by the authors in [10], providing a refined expansion of the perturbation. Our main result is the construction of a stable dynamics in the full nonradial setting for which the stationary state collapses with the universal law λ ∼ 2 e − 2 + γ 2 T − t e − ∣ ln T − t ∣ 2 where γ is the Euler constant. This improves on the earlier result by Raphael and Schweyer and gives a new robust approach to so‐called type II singularities for critical parabolic problems. A by‐product of the spectral analysis we developed is the existence of unstable blowup dynamics with speed λ ℓ ∼ C 0 T − t normalℓ 2 ln T − t − ℓ 2 () normalℓ − 1 1em for normalℓ ≥ 2 integer . © 2021 Wiley Periodicals LLC.

Collapsing-ring blowup solutions for the Keller-Segel system in three dimensions and higher

“…Finite time blow solutions had been predicted in [33], [7], [23]. Rigorous constructions were later done by Herrero-Velázquez, [19], [39], Raphaël-Schweyer [36] and the present authors [9] where the following blowup dynamics was confirmed:…”

“…This blowup dynamics is stable and is believed to be generic thanks to the partial classification result of Mizoguchi [31] who proved that (1.4) is the only blowup mechanism that occurs among radial nonnegative solutions. Other blowup rates corresponding to unstable blowup dynamics were also obtained in [9] as a consequence of a detailed spectral analysis obtained in [8].…”

“…In contrast, during type II blowup, a norm of the solution does not diverge according to the self-similar rate, which formally means that ∂ t u is subleading as t ↑ T . In the two-dimensional blowup for the Keller-Segel equation, ∂ t u is fully subleading [19,36,9], so that the profile is a stationary state. This is the most studied situation among type II blowup for evolution PDEs, see e.g.…”

Construction of a blow-up solution for a perturbed nonlinear heat equation with a gradient and a non-local term

“…This method has proved to be efficient for different PDEs from different types, and no list can be exhaustive (see del Pino, Musso and Wei [7], Nouaili and Zaag [32], Tayachi and Zaag [34], Duong and Zaag [8], Mahmoudi, Nouaili and Zaag [23], Merle, Raphael, Rodnianski and Szeftel [24], Collot, Ghoul, Nguyen and Masmoudi [6], etc...).…”

Sharp equivalent for the blowup profile to the gradient of a solution to the semilinear heat equation

“…More generally, a large literature has been devoted in the last 20 years to the construction of solutions of PDEs with prescribed behavior, beyond the case of parabolic equations such as: Type I anisotropic heat equation by Merle et al [51]; Type II blowup for heat equation by del Pino et al [21,22,23,24], Schweyer [65], Collot [13], Merle et al [12], Harada [33,34], Seki [66]; blowup for nonlinear Schrödinger equation by Merle [43], Martel and Merle [46], Merle et al [48,50,49], Raphaël and Szeftel [62]; Blowup for wave equations by Côte and Zaag [14], Ming et al [52], Collot [8], Hillairet and Raphaël [35], Krieger et al [41,40], Ghoul et al [30], Raphaël and Rodnianski [61], Donninger and Schörkhuber [25]; Blowup for KdV and gKdV [42], Côte [6,7]; Schrödinger map by Merle et al [47]; Heat flow map by Ghoul et al [31], Raphaël and Schweyer [64], Dávila et al [15]; Keller Segel system by Ghoul et al [10,11], Schweyer and Raphaël [63]; Prandtl's system by Collot et al [9]; Stefan problem by Hadzic and Raphaël [36]; 3-dimensional compressible fluids by Merle et al [49]; quenching phenomena for MEMS devices by Duong and Zaag [28]…”