[Paper Review] Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source

Cibing Yang, Xinru Cao+2 more|arXiv (Cornell University)|Mar 9, 2015

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TL;DR

This paper establishes global existence and boundedness of classical solutions to a quasilinear fully parabolic Keller-Segel system in higher dimensions ($n \geq 2$) with a logistic source term. By combining weighted $L^\gamma$ estimates, energy methods, and Moser iteration, it proves that solutions remain uniformly bounded for all time whenever $\chi/\mu < \theta_0$, where $\theta_0 > 0$ is a threshold depending on system parameters, demonstrating the critical stabilizing role of logistic damping against chemotactic blow-up.

ABSTRACT

This paper deals with the higher dimension quasilinear parabolic-parabolic Keller-Segel system involving a source term of logistic type $ u_t= abla\cdot(ϕ(u) abla u)-χ abla\cdot(u abla v)+g(u)$, $τv_t=Δv-v+u$ in $Ω imes (0,T)$, subject to nonnegative initial data and homogeneous Neumann boundary condition, where $Ω$ is smooth and bounded domain in $\mathbb{R}^n$, $n\ge 2$, $ϕ$ and $g$ are smooth and positive functions satisfying $ks^p\leϕ$ when $s\ge s_0&gt;1$, $g(s) \le as - μs^2$ for $s&gt;0$ with $g(0)\ge0$ and constants $a\ge 0$, $τ,χ,μ&gt;0$. It was known that the model without the logistic source admits both bounded and unbounded solutions, identified via the critical exponent $\frac{2}{n}$. On the other hand, the model is just a critical case with the balance of logistic damping and aggregation effects, for which the property of solutions should be determined by the coefficients involved. In the present paper it is proved that there is $θ_0&gt;0$ such that the problem admits global bounded classical solutions, regardless of the size of initial data and diffusion whenever $\fracχμ

Motivation & Objective

• To investigate the long-time behavior of solutions in a quasilinear parabolic-parabolic Keller-Segel system with logistic source in dimensions $n \geq 2$.
• To determine under what conditions the logistic source prevents finite-time blow-up despite strong chemotactic aggregation.
• To establish global existence and uniform boundedness of classical solutions independent of initial data size.
• To identify a critical threshold $\chi/\mu < \theta_0$ that ensures boundedness, highlighting the stabilizing effect of logistic damping.
• To extend previous results on critical cases where aggregation and logistic damping are balanced, particularly for $q=1$ in the chemotactic sensitivity.

Proposed method

• Derives a priori $L^\gamma$ estimates for $u$ by testing the first equation with $u^{\gamma-1}$ and substituting the second equation via variation-of-constants and energy estimates.
• Applies the variation-of-constants formula to a differential inequality involving $\int_\Omega u^\gamma$, incorporating terms from the logistic source ($-\mu u^{\gamma+1}$) and chemotactic flux ($\chi \nabla v$).
• Uses the $W^{2,\gamma+1}$-regularity of $v$ and interpolation to control $\int_\Omega |\Delta v|^{\gamma+1}$ via $\int_\Omega u^{\gamma+1}$, leveraging the parabolic smoothing effect.
• Employs Moser iteration to upgrade $L^\gamma$ boundedness to $L^\infty$ boundedness of $u$ for $\gamma \geq \gamma_0(n,p)$, ensuring uniform pointwise bounds.
• Establishes boundedness of $v$ via standard parabolic regularity theory once $u$ is bounded.
• Concludes global existence by contradiction using Lemma 2.1, under the condition $\chi/\mu < \theta_0$ with $\theta_0 = \left( \inf_{\eta > 0} \left( \eta + c_2 C_{\gamma_0+1} \eta^{-\gamma_0} \chi^{\gamma_0+1} \right) \right)^{-1}$.

Experimental results

Research questions

• RQ1 Under what conditions does the logistic source prevent blow-up in a higher-dimensional quasilinear Keller-Segel system with fully parabolic dynamics?
• RQ2 How does the balance between chemotactic aggregation ($\chi$) and logistic damping ($\mu$) affect solution boundedness?
• RQ3 Can global classical solutions exist even for large initial data when the logistic source is present?
• RQ4 What is the critical threshold $\chi/\mu < \theta_0$ that ensures boundedness, and how is it derived from the system's nonlinear structure?
• RQ5 Does the presence of nonlinear diffusion ($\phi(u) \sim u^p$) and logistic growth ($g(u) \leq a u - \mu u^2$) stabilize solutions in the critical case $q=1$?

Key findings

• There exists a threshold $\theta_0 > 0$ such that if $\chi/\mu < \theta_0$, then all classical solutions to the system remain globally bounded in time, regardless of the initial data size.
• The boundedness result holds for all $n \geq 2$, extending previous results to the fully parabolic case with nonlinear diffusion and logistic growth.
• The proof establishes $L^\gamma$ boundedness of $u$ for $\gamma \geq \gamma_0(n,p)$, which is then upgraded to $L^\infty$ boundedness via Moser iteration.
• The critical role of the logistic source is quantified: when $\chi/\mu < \theta_0$, the damping effect dominates over aggregation, preventing blow-up even in the absence of strong diffusion.
• The threshold $\theta_0$ is explicitly defined as $\theta_0 = \left( \inf_{\eta > 0} \left( \eta + c_2 C_{\gamma_0+1} \eta^{-\gamma_0} \chi^{\gamma_0+1} \right) \right)^{-1}$, showing dependence on $\chi$, $n$, $p$, and system constants.
• The boundedness of $v$ follows from standard parabolic regularity once $u$ is bounded, completing the proof of global existence and uniform boundedness of $(u,v)$.