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Blow-up solutions for the steady state of the Keller-Segel system on Riemann surfaces

Mohameden Ahmedou, Thomas Bartsch, Zhengni Hu

TL;DR

This work analyzes stationary Keller-Segel systems on compact Riemann surfaces with Neumann boundary, focusing on blow-up of solutions as $\lambda\to 4\pi m$ with $m=2k+l$. The authors implement a Lyapunov-Schmidt reduction around a multi-bubble ansatz to reduce to a finite-dimensional problem governed by the reduced energy $\mathcal{F}^V_{k,l}$, which encodes interactions via the Green's function and Robin function together with the weight $V$. They prove existence of blow-up solutions corresponding to stable critical points of $\mathcal{F}^V_{k,l}$, yielding at least $1+\lfloor m/2\rfloor$ distinct families, and they establish precise blow-up profiles and concentration behaviors in terms of the points $\xi_i$. The results extend mean-field type blow-up analysis to curved spaces with boundary, allow zeros of the weight $V$, and cover singular variants, thereby broadening the applicability to geometric and physical models with chemotaxis-like dynamics on surfaces. The approach combines variational methods, careful Green's function decompositions, isothermal coordinates, and a sharp expansion of the reduced energy to identify blow-up locations via stable critical points of $\mathcal{F}^V_{k,l}$.

Abstract

We study the following Neumann boundary problem related to the stationary solutions of the Keller-Segel system, a basic model of chemotaxis phenomena: \[ \left\{\begin{array}{ll} -Δ_g u +βu =λ\left(\frac{Ve^u}{\int_Σ Ve^u d v_g}-1\right), &\text { in } \mathringΣ\\ \partial_{ ν_g} u=0, &\text { on } \partial Σ\end{array} \right.,\] on a compact Riemann surface $(Σ, g)$ of unit area, with interior $\mathringΣ$ and smooth boundary $\partial Σ$. Here, $Δ_g$ denote the Laplace-Beltrami operator, $dv_g$ the area element of $(Σ, g)$, and $ν_g$ the unit outward normal to $\partial Σ$ and $λ$ and $β$ are non-negative parameters, $V$ is non-negative with finite zero set. For any integers $m>0$ and $k,l\geq 0$ with $m=2k+l$, we establish a sufficient condition on $V$ for the existence of a sequence of blow-up solutions as $λ$ approaches the critical values $4πm$, which blows up at $k$ points in the interior and $l$ points on the boundary. Moreover, the study expands to the corresponding singular problem.

Blow-up solutions for the steady state of the Keller-Segel system on Riemann surfaces

TL;DR

This work analyzes stationary Keller-Segel systems on compact Riemann surfaces with Neumann boundary, focusing on blow-up of solutions as with . The authors implement a Lyapunov-Schmidt reduction around a multi-bubble ansatz to reduce to a finite-dimensional problem governed by the reduced energy , which encodes interactions via the Green's function and Robin function together with the weight . They prove existence of blow-up solutions corresponding to stable critical points of , yielding at least distinct families, and they establish precise blow-up profiles and concentration behaviors in terms of the points . The results extend mean-field type blow-up analysis to curved spaces with boundary, allow zeros of the weight , and cover singular variants, thereby broadening the applicability to geometric and physical models with chemotaxis-like dynamics on surfaces. The approach combines variational methods, careful Green's function decompositions, isothermal coordinates, and a sharp expansion of the reduced energy to identify blow-up locations via stable critical points of .

Abstract

We study the following Neumann boundary problem related to the stationary solutions of the Keller-Segel system, a basic model of chemotaxis phenomena: on a compact Riemann surface of unit area, with interior and smooth boundary . Here, denote the Laplace-Beltrami operator, the area element of , and the unit outward normal to and and are non-negative parameters, is non-negative with finite zero set. For any integers and with , we establish a sufficient condition on for the existence of a sequence of blow-up solutions as approaches the critical values , which blows up at points in the interior and points on the boundary. Moreover, the study expands to the corresponding singular problem.
Paper Structure (9 sections, 21 theorems, 248 equations)

This paper contains 9 sections, 21 theorems, 248 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Lyapunov-Schmidt reduction
  4. The linearized operator
  5. The reduced functional and its expansion
  6. Proof of the Main Result
  7. Regularity theory for Neumann boundary conditions
  8. Technique estimates
  9. The partial invertibility of the linearized operator

Key Result

Theorem 1.1

Given $m\in \mathbb{N}_+, k,l \in \mathbb{N}$ with $m=2k+l$, if $K\subset\subset \Xi_{k,l}^{\prime}$ is a $C^1$-stable critical point set of $\mathcal{F}^V_{k,l}$, then there exists $\varepsilon_0>0$ such that for any $\varepsilon\in (0,\varepsilon_0)$ a family of blow-up solutions $u_{\varepsilon}$ which is convergent as measures on $\Sigma$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • proof
  • proof
  • Lemma A.1
  • proof
  • Lemma A.2: Theorem 3.2 of Wehrheim2004
  • Lemma A.3
  • Theorem A.1: Corollary 6.3 of GilbargTrudinger2001
  • ...and 29 more