Does logistic growth prevent blow-up in the fully parabolic Keller–Segel chemotaxis system in dimensions $N\ge 3$?

The fully parabolic (doubly parabolic) Keller–Segel chemotaxis system with logistic growth on a bounded domain is given by $u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\rho u-\mu u^2$, $v_t=\Delta v-v+u$ with no-flux boundary conditions, where $\chi>0$ is the chemotactic sensitivity and $\rho u-\mu u^2$ ($\mu>0$) is a logistic proliferation-and-death term. In one and two space dimensions any $\mu>0$ guarantees global bounded classical solutions for arbitrary $\chi$; for the parabolic–elliptic counterpart, in which the signal equation is elliptic, global weak solutions exist for arbitrary $\mu>0$ and global bounded classical solutions exist whenever $\mu$ exceeds an explicit threshold; in dimensions $N\ge 3$ global boundedness for the fully parabolic system is proved only when $\mu$ is sufficiently large relative to $\chi$, while blow-up is known only for parabolic–elliptic variants — with subquadratic degradation, with small quadratic damping in a mass-conserving version, or with spatially heterogeneous damping — and not for any fully parabolic system with logistic damping. The open problem, posed in the source paper of Jin, Wang and Zhang (2016) and still unsettled per the 2024–2025 literature, is to settle the dichotomy in the regime of large $\chi$ relative to $\mu$ where numerics show concentrated spike aggregation: either prove that all classical solutions are global and uniformly bounded for every $\chi>0$ and $\mu>0$, or construct parameters and initial data producing finite-time or infinite-time blow-up.

Background

Chemotaxis is the directed movement of cells or organisms along gradients of a chemical signal; it is modeled by the Keller–Segel system, in which the cell density $u(x,t)$ diffuses, grows or dies, and is additionally transported up the gradient of a chemical concentration $v(x,t)$ that the cells themselves produce. In its minimal fully parabolic form on a bounded domain $\Omega\subset\mathbb{R}^N$ with homogeneous Neumann (no-flux) boundary conditions, the model with logistic growth reads

$$\begin{cases} u_t=\Delta u-\chi\,\nabla\cdot(u\nabla v)+\rho u-\mu u^2,\\ v_t=\Delta v-v+u,\end{cases}$$

where $\chi>0$ is the chemotactic sensitivity (the strength of the directed drift term $-\chi\nabla\cdot(u\nabla v)$), $\rho\ge 0$ the intrinsic growth rate, and $\mu>0$ the coefficient of the logistic damping term $\rho u-\mu u^2$, which limits the population at the carrying capacity scale $\rho/\mu$. By a classical solution blowing up in finite time one means that the maximal classical solution ceases to exist at some $T_{\max}<\infty$ with $\limsup_{t\nearrow T_{\max}}\|u(\cdot,t)\|_{L^\infty(\Omega)}=\infty$; infinite-time blow-up means unboundedness of $\|u(\cdot,t)\|_{L^\infty(\Omega)}$ as $t\to\infty$. Without the logistic term, the chemotactic drift can overcome diffusion and cause mass concentration: in two dimensions the plain system has a critical-mass phenomenon separating global existence from chemotactic collapse (aggregation into Dirac-mass-like spikes), and in higher dimensions blow-up occurs for large data. The logistic term is therefore added both for biological realism (birth and death) and as a candidate mechanism that regularizes the model. The question of whether logistic damping always suffices became sharply posed in the pattern-formation literature: Jin, Wang and Zhang (2016) proved, via the Crandall–Rabinowitz local bifurcation theorem (a standard result asserting that curves of nontrivial solutions branch off a family of trivial ones at simple-eigenvalue crossings), that nonconstant steady states — boundary and interior spikes, stripes — emanate from the constant equilibrium as $\chi$ increases, with numerics showing stable concentrated spikes exactly when $\chi$ is large, and they listed the blow-up-prevention question as a central unresolved issue. The known state of the art is: (i) for $N\le 2$, all classical solutions are global and bounded for any $\chi$ and any constant $\mu>0$ (Osaki, Tsujikawa, Yagi and Mimura 2002, who also obtained an exponential attractor — a compact set attracting all solution trajectories at an exponential rate — and hence uniform bounds; even large classes of sub-logistic growth terms, i.e. damping weaker than quadratic, suffice per Xiang); (ii) for the parabolic–elliptic simplification, in which the second equation is replaced by the elliptic relation $0=\Delta v-v+u$ (the signal adjusts instantaneously), Tello and Winkler (2007) proved that global weak solutions exist for arbitrary $\mu>0$, and that solutions are globally classical and bounded whenever $\mu>(N-2)/N$ in the normalization $\chi=1$ — a threshold that is automatic in dimensions $N\le 2$ but a genuine largeness condition when $N\ge 3$; (iii) for the fully parabolic system in $N\ge 3$, Winkler (2010) proved global boundedness when $\mu$ is sufficiently large relative to $\chi$, with subsequent refinements by Xiang (2018), who relaxed the largeness condition in three dimensions and, in a companion paper, quantified it through explicit damping thresholds $\mu_0$ depending on the coefficients; (iv) global weak or generalized solutions exist for arbitrarily small $\mu>0$ (Lankeit 2015, with eventual smoothness in 3D under conditions; Viglialoro 2016; Winkler 2023), but this does not settle boundedness of classical solutions. Two-sided evidence keeps the question genuinely open: blow-up is known for several neighboring parabolic–elliptic variants — with subquadratic superlinear degradation (Winkler 2011 in the mass-conserving variant for $N\ge 5$; Winkler 2018 for $0=\Delta v-v+u$ in $N\ge 3$; Fuest 2021 showing the exponent $\kappa=2$ is critical in $N\ge 4$ and blow-up occurs even for pure quadratic damping with small $\mu$ in $N\ge 5$ in the mass-conserving variant; Black, Fuest and Lankeit 2021 with nonlinear diffusion), and with spatially heterogeneous damping $\mu(x)$ vanishing at a point in two dimensions (Fuest 2020) — while for fully parabolic systems no blow-up construction with logistic damping is known at all: in two dimensions with heterogeneous damping, Fuest, Lankeit and Mizukami (2024) showed only that any blow-up set must be contained in the zeros of $\mu$, and homogeneous $\mu>0$ gives global boundedness (Osaki et al. 2002). On the other side of the dichotomy, Winkler (2017) showed that in balls in dimensions $N\ge 3$ the fully parabolic system with homogeneous logistic damping develops, generically, transient densities exceeding any multiple of the carrying-capacity scale $1/\mu$ when $\chi>\mu$, so that the benign alternative is also in doubt. The source paper poses the question for its slightly more general system with a smooth positive sensitivity function $\varphi(u,v)$ in the drift term and a general smooth production rate $f(u)$ in the signal equation; the minimal system above is the standard prototype of that class and is where the entire subsequent literature locates the open problem.

Problem Statement

Consider the minimal fully parabolic Keller–Segel system with logistic growth

$$\begin{cases} u_t=\Delta u-\chi\,\nabla\cdot(u\nabla v)+\rho u-\mu u^2, & x\in\Omega,\ t>0,\\ v_t=\Delta v-v+u, & x\in\Omega,\ t>0,\\ \partial_\nu u=\partial_\nu v=0 \ \text{on}\ \partial\Omega, \end{cases}$$

on a smooth bounded domain $\Omega\subset\mathbb{R}^N$ with nonnegative sufficiently smooth initial data, where $\chi>0$, $\rho\ge 0$, $\mu>0$. Determine whether logistic damping always prevents blow-up in dimensions $N\ge 3$, the most delicate case being the regime where $\chi$ is large relative to $\mu$ in which numerical simulations show concentrated point-like spikes: either (i) prove that for every $N\ge 3$, every smooth bounded domain, every $\chi>0$ and every $\mu>0$ all classical solutions are global in time and uniformly bounded in $\Omega\times(0,\infty)$, closing the gap left by the largeness-of-$\mu$ theorems of Winkler (2010) and Xiang (2018); or (ii) exhibit some $N\ge 3$, some smooth bounded domain, some admissible parameter choice outside the known largeness-of-$\mu$ boundedness regions — any such choice qualifies, with large $\chi/\mu$, the spike-aggregation regime emphasized by the source, merely the illustrative case — and admissible initial data for which the maximal classical solution blows up in finite time ($\limsup_{t\nearrow T_{\max}}\|u(\cdot,t)\|_{L^\infty(\Omega)}=\infty$ with $T_{\max}<\infty$) or becomes unbounded as $t\to\infty$. The source paper (Jin–Wang–Zhang 2016) poses this question for its more general system with a smooth positive sensitivity function $\varphi(u,v)$ in place of $u$ in the drift term and a general smooth production rate $f(u)$ in place of $u$ in the signal equation; the minimal system above is the standard prototype of that class and is the system considered here, and the question is open already for it.

References

1. Ling Jin, Qi Wang, Zengyan Zhang (2016), 'Pattern Formation in Keller–Segel Chemotaxis Models with Logistic Growth', International Journal of Bifurcation and Chaos 26(2):1650033, DOI 10.1142/S0218127416500334 (preprint arXiv:1407.5246), https://doi.org/10.1142/S0218127416500334
2. Koichi Osaki, Tohru Tsujikawa, Atsushi Yagi, Masayasu Mimura (2002), 'Exponential attractor for a chemotaxis-growth system of equations', Nonlinear Analysis: Theory, Methods & Applications 51(1):119–144, DOI 10.1016/S0362-546X(01)00815-X, https://doi.org/10.1016/S0362-546X(01)00815-X
3. J. Ignacio Tello, Michael Winkler (2007), 'A chemotaxis system with logistic source', Communications in Partial Differential Equations 32(6):849–877, DOI 10.1080/03605300701319003, https://doi.org/10.1080/03605300701319003
4. Michael Winkler (2010), 'Boundedness in the Higher-Dimensional Parabolic-Parabolic Chemotaxis System with Logistic Source', Communications in Partial Differential Equations 35(8):1516–1537, DOI 10.1080/03605300903473426, https://doi.org/10.1080/03605300903473426
5. Michael Winkler (2011), 'Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction', Journal of Mathematical Analysis and Applications 384(2):261–272, DOI 10.1016/j.jmaa.2011.05.057, https://doi.org/10.1016/j.jmaa.2011.05.057
6. Johannes Lankeit (2015), 'Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source', Journal of Differential Equations 258(4):1158–1191, DOI 10.1016/j.jde.2014.10.016, https://doi.org/10.1016/j.jde.2014.10.016
7. Michael Winkler (2017), 'Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems', Discrete and Continuous Dynamical Systems Series B 22(7):2777–2793, DOI 10.3934/dcdsb.2017135, https://doi.org/10.3934/dcdsb.2017135
8. Michael Winkler (2018), 'Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation', Zeitschrift für angewandte Mathematik und Physik 69(2):Art. 40, DOI 10.1007/s00033-018-0935-8, https://doi.org/10.1007/s00033-018-0935-8
9. Tian Xiang (2018), 'Chemotactic aggregation versus logistic damping on boundedness in the 3D minimal Keller–Segel model', SIAM Journal on Applied Mathematics 78(5):2420–2438, DOI 10.1137/17M1150475, https://doi.org/10.1137/17M1150475
10. Tian Xiang (2018), 'Sub-logistic source can prevent blow-up in the 2D minimal Keller–Segel chemotaxis system', Journal of Mathematical Physics 59(8):081502, DOI 10.1063/1.5018861 (arXiv:1712.04739), https://doi.org/10.1063/1.5018861
11. G. Viglialoro (2016), 'Very weak global solutions to a parabolic–parabolic chemotaxis-system with logistic source', Journal of Mathematical Analysis and Applications 439(1):197–212, DOI 10.1016/j.jmaa.2016.02.069, https://doi.org/10.1016/j.jmaa.2016.02.069
12. Mario Fuest (2020), 'Finite-time blow-up in a two-dimensional Keller–Segel system with an environmental dependent logistic source', Nonlinear Analysis: Real World Applications 52:103022, DOI 10.1016/j.nonrwa.2019.103022, https://doi.org/10.1016/j.nonrwa.2019.103022
13. Mario Fuest (2021), 'Approaching optimality in blow-up results for Keller–Segel systems with logistic-type dampening', Nonlinear Differential Equations and Applications 28(2):Art. 16, DOI 10.1007/s00030-021-00677-9 (arXiv:2007.01184), https://doi.org/10.1007/s00030-021-00677-9
14. Tobias Black, Mario Fuest, Johannes Lankeit (2021), 'Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic–elliptic Keller–Segel systems', Zeitschrift für angewandte Mathematik und Physik 72(3):Art. 96, DOI 10.1007/s00033-021-01524-8 (arXiv:2005.12089), https://doi.org/10.1007/s00033-021-01524-8
15. Michael Winkler (2023), '$L^1$ solutions to parabolic Keller–Segel systems involving arbitrary superlinear degradation', Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Series V, XXIV(1):141–172, DOI 10.2422/2036-2145.202005_016, https://doi.org/10.2422/2036-2145.202005_016
16. Mario Fuest, Johannes Lankeit, Masaaki Mizukami (2025), 'Location of blow-up points in fully parabolic chemotaxis systems with spatially heterogeneous logistic source', Discrete and Continuous Dynamical Systems 45(6):1809–1822, DOI 10.3934/dcds.2024148 (preprint arXiv:2406.11746, 2024), https://doi.org/10.3934/dcds.2024148
17. Tian Xiang (2018), 'How strong a logistic damping can prevent blow-up for the minimal Keller–Segel chemotaxis system?', Journal of Mathematical Analysis and Applications 459(2):1172–1200, DOI 10.1016/j.jmaa.2017.11.022 (arXiv:1612.03024), https://doi.org/10.1016/j.jmaa.2017.11.022