Well-posedness and asymptotic limits for a degenerate Keller-Segel system with volume filling

Abstract: A class of parabolic-parabolic Keller-Segel systems with degenerate diffusion and volume filling is studied in a bounded domain subject to no-flux boundary conditions. The equations are derived from a multiphase fluid model. The interplay between nonlinear diffusion and density saturation leads to a rich variety of behaviors across different parameter regimes. We establish the existence of global weak solutions, a weak-strong uniqueness result, the exponential convergence to the homogeneous steady state, pattern formation in one spatial dimension, as well as the parabolic-elliptic and vanishing diffusion limits. The analysis relies on a priori estimates derived from suitable entropy functionals. Pattern formation is demonstrated by reducing the system to a first-order equation and conducting a detailed analysis of the resulting nonlinearity. Numerical simulations from a one-dimensional finite-volume scheme illustrate the asymptotic regimes.

• The study successfully proved existence of global weak solutions to the degenerate Keller-Segel system, including analysis at the densely filled packing threshold.
• The paper aimed for Klimov to relax to its steady state (constant or nonconstant), depending on the dimension and exponent m.
• The entropy functional framework, derived from a gradient flow structure, fully addresses well-posedness and asymptotic limits which supports the theory.

The model and its structure

The paper studies the parabolic–parabolic Keller–Segel system

$$\partial_t\rho = \operatorname{div}\big((1-\rho)\rho^{m-1}\nabla\rho - \chi\rho(1-\rho)\nabla c\big),\qquad \tau\partial_t c = \eta\Delta c - c + \rho,$$

in a bounded domain $\Omega\subset\mathbb{R}^d$ with no-flux boundary conditions. Both the diffusion coefficient $a(\rho)=(1-\rho)\rho^{m-1}$ and the chemotactic mobility $b(\rho)=\chi\rho(1-\rho)$ degenerate at the packing threshold $\rho=1$, so the system combines slow (porous-medium-type) diffusion with a volume-filling effect that caps the density at one. The authors derive the system formally from a two-phase mass–momentum balance (water plus cells) with interphase pressure $q=\rho_1^{m-1}/m$, chemotactic action–reaction forces, and linear drag; this multiphase derivation distinguishes their model from the lattice-based volume-filling models of Hillen–Painter and from the density-cutoff model of Perthame–Zhang, where $a(\rho)=\rho^{m-1}$ without the $(1-\rho)$ factor.

The system admits a gradient-flow interpretation with energy

$$E(\rho,c)=\int_\Omega\Big(\frac{\rho^m}{m(m-1)}+\frac{\chi}{2}(\eta|\nabla c|^2+c^2)-\chi\rho c\Big)dx$$

and mobility $M(\rho)=\rho(1-\rho)$, which is the structural backbone for all entropy estimates in the paper. The exponent $\Omega\subset\mathbb{R}^d$0 governs the qualitative behavior: uniqueness and relaxation to the constant state for $\Omega\subset\mathbb{R}^d$1, non-uniqueness and pattern formation for $\Omega\subset\mathbb{R}^d$2.

Global weak solutions

Existence of global weak solutions is proved for all $\Omega\subset\mathbb{R}^d$3 by an $\Omega\subset\mathbb{R}^d$4-regularization ($\Omega\subset\mathbb{R}^d$5), a Leray–Schauder fixed-point argument on the truncated problem, and uniform-in-$\Omega\subset\mathbb{R}^d$6 entropy estimates. The key technical device is the reformulation

$\Omega\subset\mathbb{R}^d$7

which yields the energy dissipation inequality controlling $\Omega\subset\mathbb{R}^d$8. Since $\Omega\subset\mathbb{R}^d$9 is not an admissible test function, the authors regularize it further via $a(\rho)=(1-\rho)\rho^{m-1}$0 and pass to the limit $a(\rho)=(1-\rho)\rho^{m-1}$1 by dominated convergence. Compactness as $a(\rho)=(1-\rho)\rho^{m-1}$2 follows from bounding $a(\rho)=(1-\rho)\rho^{m-1}$3 with $a(\rho)=(1-\rho)\rho^{m-1}$4 and applying the Aubin–Lions lemma in Moussa's monotone version. The limit solution satisfies $a(\rho)=(1-\rho)\rho^{m-1}$5 a.e., with $a(\rho)=(1-\rho)\rho^{m-1}$6 — a regularity statement adapted to the double degeneracy at both $a(\rho)=(1-\rho)\rho^{m-1}$7 and $a(\rho)=(1-\rho)\rho^{m-1}$8.

Weak–strong uniqueness

Full uniqueness of weak solutions appears out of reach: existing techniques require either $a(\rho)=(1-\rho)\rho^{m-1}$9 vanishing at $b(\rho)=\chi\rho(1-\rho)$0 or specific relations between $b(\rho)=\chi\rho(1-\rho)$1 and $b(\rho)=\chi\rho(1-\rho)$2 not satisfied here. Instead, the paper establishes weak–strong uniqueness for $b(\rho)=\chi\rho(1-\rho)$3: any weak solution coincides with a strong solution (bounded away from $b(\rho)=\chi\rho(1-\rho)$4 and $b(\rho)=\chi\rho(1-\rho)$5, with $b(\rho)=\chi\rho(1-\rho)$6) emanating from the same data. The proof uses the binary relative entropy

$b(\rho)=\chi\rho(1-\rho)$7

whose crucial feature is the lower bound $b(\rho)=\chi\rho(1-\rho)$8, stemming from $b(\rho)=\chi\rho(1-\rho)$9. After expanding the entropy production into five terms $\rho=1$0, the restriction $\rho=1$1 enters through the estimate $\rho=1$2 for $\rho=1$3, which absorbs the dominant cubic-in-$\rho=1$4 term. The assumption $\rho=1$5 is technical but necessary: it keeps the logarithms in the relative entropy well defined. Whether genuine weak–strong uniqueness extends to $\rho=1$6 remains open.

Steady states and pattern formation

Equilibrium solutions with vanishing flux reduce, where $\rho=1$7, to the scalar semilinear Neumann problem

$\rho=1$8

with $\rho=1$9 fixed by mass conservation. For $q=\rho_1^{m-1}/m$0 with $q=\rho_1^{m-1}/m$1 (and, for $q=\rho_1^{m-1}/m$2, for sufficiently small $q=\rho_1^{m-1}/m$3), the constant pair $q=\rho_1^{m-1}/m$4 is the unique steady state. The proof combines a comparison argument on $q=\rho_1^{m-1}/m$5 — absorbing the right-hand side when $q=\rho_1^{m-1}/m$6 — with a Hölder-continuity argument showing that for small $q=\rho_1^{m-1}/m$7 the density cannot touch $q=\rho_1^{m-1}/m$8 or $q=\rho_1^{m-1}/m$9: if $a(\rho)=\rho^{m-1}$0, then $a(\rho)=\rho^{m-1}$1, a contradiction once $a(\rho)=\rho^{m-1}$2.

For $a(\rho)=\rho^{m-1}$3 and $a(\rho)=\rho^{m-1}$4, the picture changes qualitatively. In one dimension, multiplying the steady equation by $a(\rho)=\rho^{m-1}$5 yields the first-order reduction

$a(\rho)=\rho^{m-1}$6

For $a(\rho)=\rho^{m-1}$7 in the interval $a(\rho)=\rho^{m-1}$8, the function $a(\rho)=\rho^{m-1}$9 has a local maximum at $(1-\rho)$0 and a local minimum at $(1-\rho)$1, and a continuity argument produces $(1-\rho)$2 with $(1-\rho)$3. Since $(1-\rho)$4 for $(1-\rho)$5 near $(1-\rho)$6 (using $(1-\rho)$7) while $(1-\rho)$8 at $(1-\rho)$9, an intermediate-value argument gives $$E(\rho,c)=\int_\Omega\Big(\frac{\rho^m}{m(m-1)}+\frac{\chi}{2}(\eta|\nabla c|^2+c^2)-\chi\rho c\Big)dx$$0 with $$E(\rho,c)=\int_\Omega\Big(\frac{\rho^m}{m(m-1)}+\frac{\chi}{2}(\eta|\nabla c|^2+c^2)-\chi\rho c\Big)dx$$1. This constructs a strictly increasing steady state, i.e., a nonconstant equilibrium, demonstrating pattern formation precisely in the regime excluded from the uniqueness theorem. The threshold structure — constant unique state for $$E(\rho,c)=\int_\Omega\Big(\frac{\rho^m}{m(m-1)}+\frac{\chi}{2}(\eta|\nabla c|^2+c^2)-\chi\rho c\Big)dx$$2, increasing states for $$E(\rho,c)=\int_\Omega\Big(\frac{\rho^m}{m(m-1)}+\frac{\chi}{2}(\eta|\nabla c|^2+c^2)-\chi\rho c\Big)dx$$3 — mirrors but strengthens the phase diagram known for the density-cutoff model, where patterns require $$E(\rho,c)=\int_\Omega\Big(\frac{\rho^m}{m(m-1)}+\frac{\chi}{2}(\eta|\nabla c|^2+c^2)-\chi\rho c\Big)dx$$4 as well; here the stronger nonlinearity $$E(\rho,c)=\int_\Omega\Big(\frac{\rho^m}{m(m-1)}+\frac{\chi}{2}(\eta|\nabla c|^2+c^2)-\chi\rho c\Big)dx$$5 complicates the analysis considerably.

Exponential convergence to equilibrium

For $$E(\rho,c)=\int_\Omega\Big(\frac{\rho^m}{m(m-1)}+\frac{\chi}{2}(\eta|\nabla c|^2+c^2)-\chi\rho c\Big)dx$$6, $$E(\rho,c)=\int_\Omega\Big(\frac{\rho^m}{m(m-1)}+\frac{\chi}{2}(\eta|\nabla c|^2+c^2)-\chi\rho c\Big)dx$$7, and $$E(\rho,c)=\int_\Omega\Big(\frac{\rho^m}{m(m-1)}+\frac{\chi}{2}(\eta|\nabla c|^2+c^2)-\chi\rho c\Big)dx$$8, solutions converge exponentially fast to the constant state:

$$E(\rho,c)=\int_\Omega\Big(\frac{\rho^m}{m(m-1)}+\frac{\chi}{2}(\eta|\nabla c|^2+c^2)-\chi\rho c\Big)dx$$9

with rate $M(\rho)=\rho(1-\rho)$0, where $M(\rho)=\rho(1-\rho)$1 is the principal Neumann eigenvalue. The proof uses the relative entropy $M(\rho)=\rho(1-\rho)$2 combining the binary entropy with $M(\rho)=\rho(1-\rho)$3, a Poincaré-type inequality for $M(\rho)=\rho(1-\rho)$4, Young's inequality tuned with $M(\rho)=\rho(1-\rho)$5 (valid since $M(\rho)=\rho(1-\rho)$6), and a logarithmic Sobolev inequality for bounded domains to close the differential inequality $M(\rho)=\rho(1-\rho)$7. The decay of $M(\rho)=\rho(1-\rho)$8 then follows by energy estimates on the $M(\rho)=\rho(1-\rho)$9-equation, giving rate $\Omega\subset\mathbb{R}^d$00. This result is consistent with the uniqueness theorem: exponential relaxation can only be asserted where the steady state is unique.

Asymptotic limits

Two singular limits are justified rigorously. Parabolic–elliptic limit ($\Omega\subset\mathbb{R}^d$01, all $\Omega\subset\mathbb{R}^d$02): the entropy functional $\Omega\subset\mathbb{R}^d$03 yields the uniform bound $\Omega\subset\mathbb{R}^d$04, together with $\Omega\subset\mathbb{R}^d$05. Consequently $\Omega\subset\mathbb{R}^d$06 strongly in $\Omega\subset\mathbb{R}^d$07 for all $\Omega\subset\mathbb{R}^d$08 and $\Omega\subset\mathbb{R}^d$09 strongly in $\Omega\subset\mathbb{R}^d$10, and the limit solves the elliptic constraint $\Omega\subset\mathbb{R}^d$11. Vanishing diffusion limit ($\Omega\subset\mathbb{R}^d$12, $\Omega\subset\mathbb{R}^d$13): the entropy $\Omega\subset\mathbb{R}^d$14 gives uniform bounds including $\Omega\subset\mathbb{R}^d$15, and Gronwall's inequality controls $\Omega\subset\mathbb{R}^d$16 on finite time intervals; the limit solves the ODE $\Omega\subset\mathbb{R}^d$17. In both cases, subsequential convergence holds, upgrading to full-sequence convergence whenever the limit problem is uniquely solvable. The main technical difficulty, as the authors note, is justifying the entropy inequalities themselves, which requires approximation arguments to handle the logarithmic terms.

Numerical experiments

A one-dimensional upwind finite-volume scheme ($\Omega\subset\mathbb{R}^d$18, $\Omega\subset\mathbb{R}^d$19, Newton iteration) illustrates the theory. With $\Omega\subset\mathbb{R}^d$20, $\Omega\subset\mathbb{R}^d$21: for $\Omega\subset\mathbb{R}^d$22 the solution converges to the constant steady state, consistent with Proposition on uniqueness, whereas for $\Omega\subset\mathbb{R}^d$23 a nonconstant steady profile emerges, indicating that the smallness condition on $\Omega\subset\mathbb{R}^d$24 in the uniqueness result is sharp in practice. Snapshots for decreasing $\Omega\subset\mathbb{R}^d$25 show convergence toward the parabolic–elliptic solution, and decreasing $\Omega\subset\mathbb{R}^d$26 shows convergence toward the reduced system with $\Omega\subset\mathbb{R}^d$27, corroborating Theorems on the asymptotic limits.

Limitations and open questions

Several restrictions qualify the results. Weak–strong uniqueness and exponential decay are confined to $\Omega\subset\mathbb{R}^d$28; the entropy method breaks down for $\Omega\subset\mathbb{R}^d$29 because the absorption identity $\Omega\subset\mathbb{R}^d$30 fails. The uniqueness theorem additionally assumes $\Omega\subset\mathbb{R}^d$31 and $\Omega\subset\mathbb{R}^d$32, and the strong solution must stay uniformly away from $\Omega\subset\mathbb{R}^d$33 and $\Omega\subset\mathbb{R}^d$34 — a condition that cannot hold globally if aggregation drives densities toward the packing threshold. Pattern formation is proved only in one spatial dimension and only asserts existence of increasing steady states, not their stability or the dynamic onset of patterning; whether such states are attractors for the evolution is not addressed. The asymptotic-limit results provide subsequential convergence only, so full convergence hinges on uniqueness of the limit problems, which is available only in restricted regimes. Finally, the derivation from the multiphase model is formal; no rigorous connection between the fluid system and the chemotaxis equation is established.

Conclusion

The paper provides a complete well-posedness framework for a doubly degenerate, volume-filling Keller–Segel system: global weak solutions for all $\Omega\subset\mathbb{R}^d$35, weak–strong uniqueness and exponential equilibration for $\Omega\subset\mathbb{R}^d$36 with $\Omega\subset\mathbb{R}^d$37, explicit construction of nonconstant increasing steady states for $\Omega\subset\mathbb{R}^d$38 in one dimension, and rigorous parabolic–elliptic and vanishing-diffusion limits. The phase transition at $\Omega\subset\mathbb{R}^d$39 between relaxation and pattern formation is the central structural finding, and the entropy-functionals toolkit — built on the gradient-flow structure with mobility $\Omega\subset\mathbb{R}^d$40 — is the unifying analytical mechanism throughout.