[Paper Review] Refined regularity analysis for a Keller-Segel-consumption system involving signal-dependent motilities

Genglin Li, Michael Winkler|arXiv (Cornell University)|Jun 27, 2022

Mathematical Biology Tumor Growth4 citations

TL;DR

This paper establishes refined regularity and eventual smoothness for solutions to a Keller-Segel consumption system with signal-dependent motility, proving global classical solutions for $ n \leq 2 $ and eventually smooth weak solutions for $ n = 3 $, both converging strongly to a semitrivial equilibrium in $ L^\infty(\Omega) $. The results extend prior existence results by showing improved regularity under stronger smoothness assumptions on $ \phi $ and initial data.

ABSTRACT

We consider the Keller-Segel-type migration-consumption system involving signal-dependent motilities, $$\left\{ \begin{array}{l} u_t = Δ\big(uϕ(v)\big), \\[1mm] v_t = Δv-uv, \end{array} ight. \qquad \qquad$$ in smoothly bounded domains $Ω\subset\mathbb{R}^n$, $n\ge 1$. Under the assumption that $ϕ\in C^1([0,\infty))$ is positive on $[0,\infty)$, and for nonnegative initial data from $(C^0(\overlineΩ))^\star imes L^\infty(Ω)$, previous literature has provided results on global existence of certain very weak solutions with possibly quite poor regularity properties, and on large time stabilization toward semitrivial equilibria with respect to the topology in $(W^{1,2}(Ω))^\star imes L^\infty(Ω)$. The present study reveals that solutions in fact enjoy significantly stronger regularity features when $0<ϕ\in C^3([0,\infty))$ and the initial data belong to $(W^{1,\infty}(Ω))^2$: It is firstly shown, namely, that then in the case $n\le 2$ an associated no-flux initial-boundary value problem even admits a global classical solution, and that each of these solutions smoothly stabilizes in the sense that as $t o\infty$ we have $$ \begin{align*} u(\cdot,t) o \frac{1}{|Ω|}\int_Ωu_0 \qquad ext{ and } \qquad v(\cdot,t) o 0 \qquad \qquad (\star) \end{align*}$$ even with respect to the norm in $L^\infty(Ω)$ in both components. In the case when $n\ge 3$, secondly, some genuine weak solutions are found to exist globally, inter alia satisfying $ abla u\in L^\frac{4}{3}_{loc}(\overlineΩ imes [0,\infty);\mathbb{R}^n)$. In the particular three-dimensional setting, any such solution is seen to become eventually smooth and to satisfy ($\star$).

Motivation & Objective

• To improve the regularity of global very weak solutions previously established for a Keller-Segel-consumption system with signal-dependent motility.
• To investigate whether stronger smoothness assumptions on $ \phi $ and initial data lead to classical or eventually smooth solutions.
• To analyze the large-time behavior of solutions, particularly their convergence to a semitrivial equilibrium in strong topologies.
• To establish that solutions become smooth and stabilize in $ L^\infty(\Omega) $ for $ n \leq 3 $, even when initial regularity is limited.
• To bridge the gap between weak solution theory and classical solution behavior in chemotaxis systems with consumption and signal-dependent diffusion.

Proposed method

• Employing a regularized approximation scheme via a parameter $ \varepsilon > 0 $, constructing solutions $ (u_\varepsilon, v_\varepsilon) $ to a perturbed version of the system.
• Deriving uniform bounds on $ u_\varepsilon $, $ v_\varepsilon $, and their gradients using energy estimates and maximal $ L^p $-regularity theory.
• Establishing convergence of approximations via weak and a.e. convergence in $ L^2_{\text{loc}} $ and $ L^1_{\text{loc}} $, leveraging compactness and stability results.
• Applying the Arzelà-Ascoli theorem to extract uniformly convergent subsequences for $ u $ and $ v $ in space-time regions with $ t > T $, ensuring eventual smoothness.
• Using interpolation and $ L^p $-bounds on $ \nabla u $, particularly $ \nabla u \in L^{4/3}_{\text{loc}}(\overline{\Omega} \times [0,\infty); \mathbb{R}^n) $, to infer higher integrability and regularity.
• Leveraging auxiliary differential inequalities (e.g., Lemma 5.3) to control growth and derive uniform bounds on solution components, especially in the three-dimensional case.

Experimental results

Research questions

• RQ1 Under what conditions on $ \phi $ and initial data does the Keller-Segel-consumption system with signal-dependent motility admit global classical solutions?
• RQ2 Can solutions that are initially only very weak become eventually smooth, and if so, under what regularity assumptions on $ \phi $?
• RQ3 Do solutions converge to the semitrivial equilibrium $ (\frac{1}{|\Omega|}\int_\Omega u_0, 0) $ in the strong $ L^\infty(\Omega) $-topology?
• RQ4 What is the role of spatial dimension $ n $ in determining the regularity and long-time behavior of solutions?
• RQ5 Can the structure of the system, particularly the signal consumption term $ -uv $, prevent blow-up and enforce stabilization even with nonlinear diffusion?

Key findings

• For $ n \leq 2 $, if $ \phi \in C^3([0,\infty)) $ and initial data $ (u_0, v_0) \in (W^{1,\infty}(\Omega))^2 $, the system admits a global classical solution.
• In the case $ n \leq 2 $, the classical solution satisfies $ u(\cdot,t) \to \frac{1}{|\Omega|}\int_\Omega u_0 $ and $ v(\cdot,t) \to 0 $ in $ L^\infty(\Omega) $ as $ t \to \infty $.
• For $ n \geq 3 $, particularly $ n = 3 $, the system admits global weak solutions satisfying $ \nabla u \in L^{4/3}_{\text{loc}}(\overline{\Omega} \times [0,\infty); \mathbb{R}^n) $.
• In the three-dimensional setting, any such weak solution becomes eventually smooth and stabilizes strongly in $ L^\infty(\Omega) $ toward the semitrivial equilibrium.
• The convergence to equilibrium is strong in both components, even under minimal assumptions on $ \phi $ beyond $ C^3 $-smoothness and positivity.
• The results demonstrate that signal consumption and signal-dependent motility jointly suppress blow-up and enforce long-term stabilization, even in higher dimensions, under mild regularity conditions.