[Paper Review] Boundary spike-layer solutions of the singular Keller-Segel system: existence and stability

José A. Carrillo, Jingyu Li+1 more|arXiv (Cornell University)|Aug 20, 2019

Mathematical Biology Tumor Growth36 references4 citations

TL;DR

This paper establishes the existence and nonlinear stability of boundary spike-layer solutions for the singular Keller-Segel system with logarithmic sensitivity in the half-space, using a novel Cole-Hopf-type transformation to handle the logarithmic singularity and proving asymptotic convergence via weighted energy estimates and Hardy’s inequality. The key result is the first global well-posedness and stability analysis for the system with general consumption rates $ m \geq 0 $, including $ m \neq 1 $.

ABSTRACT

We exploit the existence and nonlinear stability of boundary spike/layer solutions of the Keller-Segel system with logarithmic singular sensitivity in the half space, where the physical zero-flux and Dirichlet boundary conditions are prescribed. We first prove that, under above boundary conditions, the Keller-Segel system admits a unique boundary spike-layer steady state where the first solution component (bacterial density) of the system concentrates at the boundary as a Dirac mass and the second solution component (chemical concentration) forms a boundary layer profile near the boundary as the chemical diffusion coefficient tends to zero. Then we show that this boundary spike-layer steady state is asymptotically nonlinearly stable under appropriate perturbations. As far as we know, this is the first result obtained on the global well-posedness of the singular Keller-Segel system with nonlinear consumption rate. We introduce a novel strategy of relegating the singularity, via a Cole-Hopf type transformation, to a nonlinear nonlocality which is resolved by the technique of "taking antiderivatives", i.e. working at the level of the distribution function. Then, we carefully choose weight functions to prove our main results by suitable weighted energy estimates with Hardy's inequality that fully captures the dissipative structure of the system.

Motivation & Objective

• To establish the existence of unique boundary spike-layer steady states in the singular Keller-Segel system with logarithmic sensitivity in the half-space.
• To prove the nonlinear asymptotic stability of these boundary spike-layer solutions under zero-flux and non-homogeneous Dirichlet boundary conditions.
• To resolve the analytical challenges posed by the logarithmic singularity for general $ m \geq 0 $, especially when $ m \neq 1 $, where prior methods fail.
• To develop a new strategy to transform the singularity into a nonlocal nonlinearity via a Cole-Hopf-type transformation, enabling the use of antiderivative techniques.
• To prove global well-posedness and large-time behavior of solutions using weighted energy estimates with Hardy’s inequality, capturing the system’s dissipative structure.

Proposed method

• Apply a Cole-Hopf-type transformation to relocate the logarithmic singularity into a nonlinear nonlocal term, simplifying the system’s structure.
• Work at the level of the distribution function (i.e., use antiderivatives) to handle the transformed system and recover regularity.
• Introduce carefully chosen weight functions in energy estimates to control the singular behavior and exploit the dissipative nature of the system.
• Employ weighted $ L^2 $ energy estimates combined with Hardy’s inequality to bound the decay of perturbations in $ u $ and $ w $.
• Use Taylor expansion and Young’s inequality to control nonlinear terms in the perturbation equations for $ w - W $.
• Establish uniform bounds on $ \|\phi\|_{L^2} $, $ \|\psi\|_{L^2} $, and their derivatives, leading to convergence to the steady state.

Experimental results

Research questions

• RQ1 Does the singular Keller-Segel system with logarithmic sensitivity and general consumption rate $ m \geq 0 $ admit a unique boundary spike-layer steady state in the half-space?
• RQ2 Can the nonlinear stability of such a boundary spike-layer solution be established under zero-flux and non-homogeneous Dirichlet boundary conditions?
• RQ3 Is it possible to overcome the analytical barriers introduced by the logarithmic singularity when $ m \neq 1 $, where standard Cole-Hopf methods fail?
• RQ4 How can the dissipative structure of the system be fully captured in the presence of boundary layers and singular sensitivity?
• RQ5 What novel transformation and energy estimate techniques are required to prove global well-posedness and large-time convergence for this system?

Key findings

• The system admits a unique non-constant steady state $ (U, W) $ where $ u $ concentrates as a Dirac mass at the boundary and $ w $ forms a boundary layer as $ \varepsilon \to 0 $.
• The boundary spike-layer steady state is asymptotically nonlinearly stable: perturbations in $ u $ and $ w $ decay to zero in $ L^1 $ and $ L^\infty $ norms as $ t \to \infty $.
• The $ L^1 $ convergence of $ \phi = u - U $ is proven via $ \int_0^\infty \frac{\phi_x^2}{U} \to 0 $, which implies $ \|\phi_x\|_{L^1} \to 0 $, using Hölder’s inequality and integrability of $ U $.
• For the chemical component, $ \xi = w - W $ satisfies $ \|\xi\|_{L^2} \to 0 $ and $ \|\xi_x\|_{L^2} \to 0 $, with uniform bounds depending only on initial data.
• The $ L^\infty $ norm of $ \xi $ decays to zero as $ t \to \infty $, due to Cauchy-Schwarz and the decay of $ \|\xi_x\|_{L^2} $, proving pointwise convergence.
• The method is robust for both $ 0 \leq m < 1 $ and $ m \geq 1 $, with different weight functions $ w_i, w_j $ chosen accordingly in the energy estimates.