For the fully parabolic (parabolic–parabolic) Keller–Segel system on the whole plane, the sharp role of the conserved cell mass is settled only on the global-existence side: solutions with mass M<8π/βM<8\pi/\beta exist globally (Calvez–Corrias, 2008), and global existence at the critical mass M=8π/βM=8\pi/\beta is known under symmetry or moment conditions. Whether supercritical mass M>8π/βM>8\pi/\beta forces finite-time blow-up — the exact analogue of the classical parabolic–elliptic dichotomy — remains open for general (non–radially symmetric) data. Radially symmetric blow-up above a threshold at least 8π8\pi (and growing with the chemical's relaxation time) is known. The problem is to prove or refute supercritical-mass blow-up for general data, completing the mass-threshold dichotomy for the original chemotaxis model.

Background

The Keller–Segel system is the classical partial-differential-equation model of chemotaxis, the directed motion of cells or bacteria up gradients of a chemoattractant that they themselves secrete; the resulting positive feedback can drive macroscopic aggregation. In its minimal two-dimensional form the cell density n(t,x)n(t,x) and chemoattractant concentration v(t,x)v(t,x) satisfy

∂tn=Δn−∇⋅(n∇v),ε ∂tv=Δv−γv+βn,(t,x)∈(0,∞)×R2,\partial_t n = \Delta n - \nabla\cdot(n\nabla v), \qquad \varepsilon\,\partial_t v = \Delta v - \gamma v + \beta n, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^2,

where γ≥0\gamma\ge 0 is the chemical's degradation rate, β>0\beta>0 its production rate, and ε∈{0,1}\varepsilon\in\{0,1\} distinguishes the parabolic–elliptic simplification (ε=0\varepsilon=0, chemical in quasi-static equilibrium) from the fully parabolic system (ε=1\varepsilon=1, chemical with its own diffusion). The total mass M=∫R2n0 dxM=\int_{\mathbb{R}^2}n_0\,dx is conserved, and blow-up means ∥n(t)∥L∞→∞\|n(t)\|_{L^\infty}\to\infty as tt approaches some finite time — biologically, chemotactic collapse onto a point.

In two dimensions the mass alone decides the fate of solutions in the parabolic–elliptic case: for M<8π/βM<8\pi/\beta all solutions exist globally, while for M>8π/βM>8\pi/\beta every solution with finite second moment ∫∣x∣2n0 dx<∞\int|x|^2 n_0\,dx<\infty blows up in finite time, a dichotomy Calvez–Corrias extended to γ≥0\gamma\ge 0 (their Theorem 1.2, with a smallness condition on the γ\gamma-weighted moment). The threshold value 8π/β8\pi/\beta reflects the sharp constant in the logarithmic Hardy–Littlewood–Sobolev inequality, an integral inequality bounding the logarithmic attraction energy in terms of the mass; the blow-up proof rests on a virial identity — the second moment ∫∣x∣2n dx\int|x|^2 n\,dx satisfies an explicit differential equation whose right-hand side becomes strictly decreasing once M>8π/βM>8\pi/\beta, so the moment would turn negative unless the solution blows up first.

For the fully parabolic system ε=1\varepsilon=1, Calvez–Corrias (2008) proved global existence for subcritical mass M<8π/βM<8\pi/\beta (allowing any γ≥0\gamma\ge 0, under a logarithmic moment condition), using a Lyapunov functional — a decreasing free energy combining cell diffusion with attraction energy — together with a Trudinger–Moser-type inequality, a sharp embedding on the plane controlling the attraction energy below the threshold mass. They state that the corresponding supercritical blow-up result for ε=1\varepsilon=1 on the plane is, to their knowledge, still an open problem. The obstruction is structural: because vv has its own time evolution, the virial identity for ∫∣x∣2n dx\int|x|^2 n\,dx acquires extra terms from the chemical's finite relaxation time that no known identity controls, and the energy dissipation couples nn and vv.

What is known on the supercritical side is essentially radial: Herrero–Velázquez (1997) constructed radially symmetric finite-time collapse for the doubly parabolic system in a ball with Neumann boundary conditions, describing the asymptotics near the singularity; Mizoguchi (2020) proved that whole-plane radially symmetric solutions blow up in finite time whenever the mass exceeds a threshold M(τ)≥max⁡{8π,Cτ}M(\tau)\ge\max\{8\pi, C\tau\} that grows with the chemical's relaxation time τ\tau (the parameter ε\varepsilon), and that any blowing-up non-symmetric solution is of type II, i.e. its blow-up rate is faster than the scaling rate (T−t)−1(T-t)^{-1}. Mizoguchi (2020) also gave initial-energy blow-up criteria in balls for space dimensions two through four. On the global-existence side, the critical mass M=8πM=8\pi was handled under radial symmetry or a logarithmic moment condition, and a 2026 preprint of Hosono extends global existence to general critical-mass data on the plane. Ferreira–Postigo (2019) likewise leave the Calvez–Corrias supercritical blow-up question open.

Problem Statement

Consider the two-dimensional fully parabolic Keller–Segel system on the whole plane, written after rescaling β=1\beta=1:

∂tn=Δn−∇⋅(n∇v),∂tv=Δv−γv+n,(t,x)∈(0,∞)×R2,γ≥0,\partial_t n = \Delta n - \nabla\cdot(n\nabla v), \qquad \partial_t v = \Delta v - \gamma v + n, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^2, \qquad \gamma\ge 0,

with smooth nonnegative initial data (n0,v0)(n_0,v_0) such that v0∈H1(R2)v_0\in H^1(\mathbb{R}^2), so that the Cauchy problem is locally well-posed, where n0n_0 has finite supercritical mass M=∫R2n0 dx>8πM=\int_{\mathbb{R}^2}n_0\,dx>8\pi and finite second moment ∫R2∣x∣2n0 dx<∞\int_{\mathbb{R}^2}|x|^2 n_0\,dx<\infty; the mass MM is conserved along solutions. Prove that every such classical solution blows up in finite time, i.e. ∥n(t)∥L∞→∞\|n(t)\|_{L^\infty}\to\infty as t↑Tt\uparrow T for some T<∞T<\infty, or refute this by constructing a smooth nonnegative solution with M>8πM>8\pi and finite second moment that exists globally in time. A resolution would establish for the fully parabolic system the same sharp critical-mass threshold 8π8\pi that is known for the parabolic–elliptic system. The still-open core is general (non–radially symmetric) data: radially symmetric blow-up above a threshold at least 8π8\pi and growing with the chemical's relaxation time is already known, as is global existence for M<8πM<8\pi (and at the critical mass M=8πM=8\pi under radial symmetry or a logarithmic moment condition, with general critical data settled so far only in a 2026 preprint).

References

  1. Lucas C. F. Ferreira and Monisse Postigo, Global well-posedness and asymptotic behavior in Besov-Morrey spaces for chemotaxis-Navier-Stokes fluids, Journal of Mathematical Physics 60 (2019); DOI 10.1063/1.5080248; arXiv:1811.02709, https://arxiv.org/abs/1811.02709
  2. V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in R^2, Communications in Mathematical Sciences 6 (2008), no. 2, 417-447; DOI 10.4310/cms.2008.v6.n2.a8; arXiv:0712.3169, https://arxiv.org/abs/0712.3169
  3. Miguel A. Herrero and Juan J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (4) 24 (1997), no. 4, 633-683; Zbl 0904.35037, https://numdam.org/item/ASNSP_1997_4_24_4_633_0/
  4. Noriko Mizoguchi, Finite-time blowup in Cauchy problem of parabolic-parabolic chemotaxis system, Journal de Mathématiques Pures et Appliquées (9) 136 (2020), 203-238; DOI 10.1016/j.matpur.2019.10.004, https://doi.org/10.1016/j.matpur.2019.10.004
  5. Noriko Mizoguchi, Criterion on initial energy for finite-time blowup in parabolic-parabolic Keller-Segel system, SIAM Journal on Mathematical Analysis 52 (2020), no. 6, 5840-5864; DOI 10.1137/19m1280570, https://doi.org/10.1137/19m1280570
  6. Tatsuya Hosono, Global existence for the fully parabolic Keller--Segel system with critical mass on the plane, preprint (2026); arXiv:2602.03768, https://arxiv.org/abs/2602.03768