Refined Asymptotic Behavior of Blowup Solutions to a Simplified Chemotaxis System

Mizoguchi, Noriko

doi:10.1002/cpa.21954

Cited by 14 publications

(6 citation statements)

References 31 publications

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“…In particular, in [ 12 ] it is shown that the blowup solution corresponding to for a certain explicit constant , is stable outside of radial symmetry. In addition to this, Mizoguchi [ 39 ] recently proved that for solutions with non-negative and radial initial data, ( 1.3 )–( 1.4 ) describes the universal blowup mechanism.…”

Section: Introductionmentioning

confidence: 97%

Stable Singularity Formation for the Keller–Segel System in Three Dimensions

Glogić,

Schörkhuber

2024

Arch Rational Mech Anal

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We consider the parabolic–elliptic Keller–Segel system in dimensions $$d \geqq 3$$ d ≧ 3 , which is the mass supercritical case. This system is known to exhibit rich dynamical behavior including singularity formation via self-similar solutions. An explicit example was found more than two decades ago by Brenner et al. (Nonlinearity 12(4):1071–1098, 1999), and is conjectured to be nonlinearly radially stable. We prove this conjecture for $$d=3$$ d = 3 . Our approach consists of reformulating the problem in similarity variables and studying the Cauchy evolution in intersection Sobolev spaces via semigroup theory methods. To solve the underlying spectral problem, we use a technique we recently established in Glogić and Schörkhuber (Comm Part Differ Equ 45(8):887–912, 2020). To the best of our knowledge, this provides the first result on stable self-similar blowup for the Keller–Segel system. Furthermore, the extension of our result to any higher dimension is straightforward. We point out that our approach is general and robust, and can therefore be applied to a wide class of parabolic models.

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Section: Introductionmentioning

confidence: 97%

Stable Singularity Formation for the Keller–Segel System in Three Dimensions

Glogić,

Schörkhuber

2024

Arch Rational Mech Anal

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“…where U (x) = 8(1 + |x| 2 ) −2 is stationary and satisfies R 2 U (x)dx = 8π. 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].…”

Section: 1) {3dsys} {3dsys}mentioning

confidence: 92%

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

Collot¹,

Ghoul²,

Masmoudi³

et al. 2021

Preprint

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We consider the parabolic-elliptic Keller-Segel system in three dimensions and higher, corresponding to the mass supercritical case. We construct rigorously a solution which blows up in finite time by having its mass concentrating near a ring that shrinks to a point. In particular, the singularity is of type II, non self-similar. We show the stability of this dynamics among spherically symmetric solutions. In renormalised variables, the solution ressembles a traveling wave imploding at the origin, and this, to our knowledge, is the first stability result for such phenomenon for an evolution PDE. We develop a framework to handle the interactions between the two blowup zones contributing to the mechanism: a thin inner zone around the ring where viscosity effects occur, and an outer zone where the evolution is mostly inviscid. lim sup t→T u(t) L ∞ (R d ) = +∞.

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“…The solutions are globally bounded if ´Ω u 0 dx < 4π (or ´Ω u 0 dx < 8π in the radial setting). For more related results, we refer to [7,17,24,26,38] and the references therein.…”

Section: Blow-up Versus Global Existence For Classical Keller-segel S...mentioning

confidence: 99%

Finite-time blow-up and boundedness in a 2D Keller–Segel system with rotation

Wang

2022

Nonlinearity

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This paper deals with the initial-boundary value problem for a Keller–Segel system with rotation { u t = Δ u − ∇ ⋅ ( u S θ ∇ v ) , x ∈ Ω , t > 0 , 0 = Δ v − v + u , x ∈ Ω , t > 0 , with zero-flux boundary condition for u and zero-Neumann boundary condition for v, where Ω is a bounded domain in R 2 with smooth boundary ∂ Ω , S θ = [ cos θ − sin θ sin θ cos θ ] , is a rotation matrix with θ ∈ ( − π 2 , π 2 ) . We show that: Let Ω ⊂ R 2 be a general smooth bounded domain. If m > 8 π cos θ , then there exists nonnegative initial data u 0 satisfying ∫ Ω u 0 d x = m , such that the corresponding nonradial solution of system (*) blows up in finite time and the blow-up point lies in Ω. If m > 4 π cos θ and ∂ Ω contains a line segment, then there exists nonnegative initial data u 0 satisfying ∫ Ω u 0 d x = m , such that the corresponding nonradial solution of system (*) blows up in finite time and the blow-up point lies on the line segment of ∂ Ω . Let Ω = B R ( 0 ) be a disc in R 2 with radius R > 0 centered at origin. Although there is a rotation effect in system (*), solutions still preserve radial symmetry of initial data. If nonnegative radially symmetric initial data u 0 satisfies ∫ Ω u 0 d x < 8 π cos θ , then the corresponding radial solution of system (*) exists globally in time and is globally bounded.

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Refined Asymptotic Behavior of Blowup Solutions to a Simplified Chemotaxis System

Cited by 14 publications

References 31 publications

Stable Singularity Formation for the Keller–Segel System in Three Dimensions

Stable Singularity Formation for the Keller–Segel System in Three Dimensions

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

Finite-time blow-up and boundedness in a 2D Keller–Segel system with rotation

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