Quasilinear Two-Species Chemotaxis

• The topic investigates a quasilinear two-species chemotaxis system characterized by nonlinear diffusion and chemotactic cross-coupling with power-law dependencies.
• It establishes critical thresholds in the (p, q, n) parameter space that separate finite-time blow-up, global boundedness, and global existence regimes.
• The model provides insights into biological pattern formation and aggregation using analytical methods such as subsolution techniques and Lyapunov functional analysis.

A quasilinear two-species chemotaxis system describes the evolution of two interacting populations whose spatial dispersal is governed by nonlinear diffusion and chemotactic cross-coupling, with each species both producing and sensing distinct chemical signals. These systems model multi-population pattern formation, aggregation, and singularity formation in biological and physical contexts. The mathematical structure is distinguished by power-law or otherwise nonlinear dependence of the diffusion and chemotactic sensitivity functions on the species densities—yielding sharp thresholds that delineate regimes of finite-time blow-up, global boundedness, and long-time existence.

1. Mathematical Formulation and Model Architecture

The canonical quasilinear two-species, two-chemical system, as introduced in Zeng & Li (Zeng et al., 8 Jan 2026), is formulated on a smooth bounded domain $\Omega \subset \mathbb{R}^n$ ($n \geq 3$) with homogeneous Neumann boundary conditions: $$\begin{cases} u_t = \nabla \cdot(D(u)\nabla u) - \nabla \cdot (S(u) \nabla v), & x \in \Omega, \ t > 0,\ 0 = \Delta v - \mu_w + w, \quad \mu_w = |\Omega|^{-1} \int_\Omega w, & x \in \Omega,\ w_t = \Delta w - \nabla \cdot(w \nabla z), & x \in \Omega,\ 0 = \Delta z - \mu_u + u, \quad \mu_u = |\Omega|^{-1} \int_\Omega u, & x \in \Omega,\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = \frac{\partial z}{\partial \nu} = 0, & x \in \partial \Omega,\ u(x,0) = u_0(x), \quad w(x,0) = w_0(x), & x \in \Omega. \end{cases}$$ Here, $u$ and $w$ denote the densities of two cell species. The potentials $v$ and $z$ are chemoattractants produced/sensed in an interlaced (cross-coupling) fashion: $u$ senses $v$ produced by $w$, and $n \geq 3$0 senses $n \geq 3$1 produced by $n \geq 3$2. The functions $n \geq 3$3 and $n \geq 3$4 capture the nonlinear (possibly degenerate or singular) diffusivity and chemotactic sensitivity for large $n \geq 3$5: $n \geq 3$6 This quasilinear structure markedly influences the regularity, aggregation, and blow-up properties of the system.

2. Critical Exponents and Sharp Blow-Up Thresholds

A defining feature of this system is the existence of critical lines in the $n \geq 3$7 parameter space, separating qualitatively distinct dynamical regimes. For the model above (Zeng et al., 8 Jan 2026):

• Finite-Time Blow-Up (FTBU): On the ball $n \geq 3$8, if $n \geq 3$9 and $$\begin{cases} u_t = \nabla \cdot(D(u)\nabla u) - \nabla \cdot (S(u) \nabla v), & x \in \Omega, \ t > 0,\ 0 = \Delta v - \mu_w + w, \quad \mu_w = |\Omega|^{-1} \int_\Omega w, & x \in \Omega,\ w_t = \Delta w - \nabla \cdot(w \nabla z), & x \in \Omega,\ 0 = \Delta z - \mu_u + u, \quad \mu_u = |\Omega|^{-1} \int_\Omega u, & x \in \Omega,\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = \frac{\partial z}{\partial \nu} = 0, & x \in \partial \Omega,\ u(x,0) = u_0(x), \quad w(x,0) = w_0(x), & x \in \Omega. \end{cases}$$0, there exist radially symmetric initial data for which $$\begin{cases} u_t = \nabla \cdot(D(u)\nabla u) - \nabla \cdot (S(u) \nabla v), & x \in \Omega, \ t > 0,\ 0 = \Delta v - \mu_w + w, \quad \mu_w = |\Omega|^{-1} \int_\Omega w, & x \in \Omega,\ w_t = \Delta w - \nabla \cdot(w \nabla z), & x \in \Omega,\ 0 = \Delta z - \mu_u + u, \quad \mu_u = |\Omega|^{-1} \int_\Omega u, & x \in \Omega,\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = \frac{\partial z}{\partial \nu} = 0, & x \in \partial \Omega,\ u(x,0) = u_0(x), \quad w(x,0) = w_0(x), & x \in \Omega. \end{cases}$$1 blow up in finite time.
• Global Boundedness (GB): On any smooth $$\begin{cases} u_t = \nabla \cdot(D(u)\nabla u) - \nabla \cdot (S(u) \nabla v), & x \in \Omega, \ t > 0,\ 0 = \Delta v - \mu_w + w, \quad \mu_w = |\Omega|^{-1} \int_\Omega w, & x \in \Omega,\ w_t = \Delta w - \nabla \cdot(w \nabla z), & x \in \Omega,\ 0 = \Delta z - \mu_u + u, \quad \mu_u = |\Omega|^{-1} \int_\Omega u, & x \in \Omega,\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = \frac{\partial z}{\partial \nu} = 0, & x \in \partial \Omega,\ u(x,0) = u_0(x), \quad w(x,0) = w_0(x), & x \in \Omega. \end{cases}$$2, for $$\begin{cases} u_t = \nabla \cdot(D(u)\nabla u) - \nabla \cdot (S(u) \nabla v), & x \in \Omega, \ t > 0,\ 0 = \Delta v - \mu_w + w, \quad \mu_w = |\Omega|^{-1} \int_\Omega w, & x \in \Omega,\ w_t = \Delta w - \nabla \cdot(w \nabla z), & x \in \Omega,\ 0 = \Delta z - \mu_u + u, \quad \mu_u = |\Omega|^{-1} \int_\Omega u, & x \in \Omega,\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = \frac{\partial z}{\partial \nu} = 0, & x \in \partial \Omega,\ u(x,0) = u_0(x), \quad w(x,0) = w_0(x), & x \in \Omega. \end{cases}$$3, all classical solutions remain globally bounded.
• Global Existence (GE): On any smooth $$\begin{cases} u_t = \nabla \cdot(D(u)\nabla u) - \nabla \cdot (S(u) \nabla v), & x \in \Omega, \ t > 0,\ 0 = \Delta v - \mu_w + w, \quad \mu_w = |\Omega|^{-1} \int_\Omega w, & x \in \Omega,\ w_t = \Delta w - \nabla \cdot(w \nabla z), & x \in \Omega,\ 0 = \Delta z - \mu_u + u, \quad \mu_u = |\Omega|^{-1} \int_\Omega u, & x \in \Omega,\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = \frac{\partial z}{\partial \nu} = 0, & x \in \partial \Omega,\ u(x,0) = u_0(x), \quad w(x,0) = w_0(x), & x \in \Omega. \end{cases}$$4, if $$\begin{cases} u_t = \nabla \cdot(D(u)\nabla u) - \nabla \cdot (S(u) \nabla v), & x \in \Omega, \ t > 0,\ 0 = \Delta v - \mu_w + w, \quad \mu_w = |\Omega|^{-1} \int_\Omega w, & x \in \Omega,\ w_t = \Delta w - \nabla \cdot(w \nabla z), & x \in \Omega,\ 0 = \Delta z - \mu_u + u, \quad \mu_u = |\Omega|^{-1} \int_\Omega u, & x \in \Omega,\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = \frac{\partial z}{\partial \nu} = 0, & x \in \partial \Omega,\ u(x,0) = u_0(x), \quad w(x,0) = w_0(x), & x \in \Omega. \end{cases}$$5, all classical solutions exist globally (without necessarily being bounded).

Thus, two specific lines,

$$\begin{cases} u_t = \nabla \cdot(D(u)\nabla u) - \nabla \cdot (S(u) \nabla v), & x \in \Omega, \ t > 0,\ 0 = \Delta v - \mu_w + w, \quad \mu_w = |\Omega|^{-1} \int_\Omega w, & x \in \Omega,\ w_t = \Delta w - \nabla \cdot(w \nabla z), & x \in \Omega,\ 0 = \Delta z - \mu_u + u, \quad \mu_u = |\Omega|^{-1} \int_\Omega u, & x \in \Omega,\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = \frac{\partial z}{\partial \nu} = 0, & x \in \partial \Omega,\ u(x,0) = u_0(x), \quad w(x,0) = w_0(x), & x \in \Omega. \end{cases}$$6

partition the phase space into three regimes (FTBU, GB, GE). The balance $$\begin{cases} u_t = \nabla \cdot(D(u)\nabla u) - \nabla \cdot (S(u) \nabla v), & x \in \Omega, \ t > 0,\ 0 = \Delta v - \mu_w + w, \quad \mu_w = |\Omega|^{-1} \int_\Omega w, & x \in \Omega,\ w_t = \Delta w - \nabla \cdot(w \nabla z), & x \in \Omega,\ 0 = \Delta z - \mu_u + u, \quad \mu_u = |\Omega|^{-1} \int_\Omega u, & x \in \Omega,\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = \frac{\partial z}{\partial \nu} = 0, & x \in \partial \Omega,\ u(x,0) = u_0(x), \quad w(x,0) = w_0(x), & x \in \Omega. \end{cases}$$7 formalizes the interplay between nonlinear chemotactic aggregation and nonlinear diffusion: if chemotactic sensitivity (for large densities) outpaces diffusion beyond the critical line, singularity formation occurs. If diffusion is sufficiently dominant, aggregation is bounded. For sufficiently weak chemosensitivity ($$\begin{cases} u_t = \nabla \cdot(D(u)\nabla u) - \nabla \cdot (S(u) \nabla v), & x \in \Omega, \ t > 0,\ 0 = \Delta v - \mu_w + w, \quad \mu_w = |\Omega|^{-1} \int_\Omega w, & x \in \Omega,\ w_t = \Delta w - \nabla \cdot(w \nabla z), & x \in \Omega,\ 0 = \Delta z - \mu_u + u, \quad \mu_u = |\Omega|^{-1} \int_\Omega u, & x \in \Omega,\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = \frac{\partial z}{\partial \nu} = 0, & x \in \partial \Omega,\ u(x,0) = u_0(x), \quad w(x,0) = w_0(x), & x \in \Omega. \end{cases}$$8 subcritical), global existence is ensured regardless of diffusion degeneracy.

Theoretical analysis for classical and related systems with different nonlinearities, such as the flux-limited variant and power-law diffusions, confirms analogous sharp-phase separation, often with threshold curves or surfaces in the parameter space for $$\begin{cases} u_t = \nabla \cdot(D(u)\nabla u) - \nabla \cdot (S(u) \nabla v), & x \in \Omega, \ t > 0,\ 0 = \Delta v - \mu_w + w, \quad \mu_w = |\Omega|^{-1} \int_\Omega w, & x \in \Omega,\ w_t = \Delta w - \nabla \cdot(w \nabla z), & x \in \Omega,\ 0 = \Delta z - \mu_u + u, \quad \mu_u = |\Omega|^{-1} \int_\Omega u, & x \in \Omega,\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = \frac{\partial z}{\partial \nu} = 0, & x \in \partial \Omega,\ u(x,0) = u_0(x), \quad w(x,0) = w_0(x), & x \in \Omega. \end{cases}$$9 (Zeng et al., 8 Jan 2026, Cai et al., 8 Jan 2026).

3. Analytical Methods and Proof Strategies

Blow-Up Construction

The blow-up regime is established via transformation to mass-distributions $u$0 (cumulative radial densities). Subsolution techniques are employed: explicit lower solutions with suitably constructed scaling profiles $u$1 produce finite-time gradient blow-up at the origin. Algebraic inequalities involving $u$2 arise from ensuring these subsolutions are dominated by the actual solution, yielding the sharp FTBU conditions (Zeng et al., 8 Jan 2026).

Lyapunov and Entropy Methods

For boundedness, construction and dissipation analysis of an energy/entropy functional is central: $u$3 Lyapunov monotonicity ($u$4) and suitable a priori estimates for $u$5 and $u$6 (via Moser-iteration, elliptic regularity) in the regime $u$7 yield $u$8-bounds that preclude blow-up (Zeng et al., 8 Jan 2026).

Elementary Estimates for Subcritical Chemosensitivity

In the regime $u$9, straightforward $w$0-type estimates combined with Sobolev embedding suffice for global existence without recourse to a Lyapunov functional, as chemosensitivity is inherently too weak to generate singularity.

A summary of proof techniques for related variants is given in (Zeng et al., 8 Jan 2026, Cai et al., 8 Jan 2026), including the role of flux limitation and coupled mass-distribution ODEs, which provide alternative critical exponents.

4. Biological Interpretation of Nonlinearities

The exponents $w$1 and $w$2 have concrete biological interpretations:

• $w$3 quantifies the nonlinearity in diffusion $w$4: higher $w$5 corresponds to stronger "flattening" or crowding avoidance at high densities.
• $w$6 represents the nonlinearity in chemosensitivity $w$7: larger $w$8 amplifies aggregation in high-density regimes.

The parameter $w$9 thus operationally measures whether chemotactic cross-attraction dominates diffusive dispersal in the macroscopic biological model. Biological systems with $v$0 beyond the critical threshold can display spontaneous self-organization into singular aggregates, while lower $v$1 promotes pattern formation without singularity. For sufficiently small $v$2, even strongly degenerate or singularly nonlinear diffusion does not permit catastrophic collapse (Zeng et al., 8 Jan 2026).

5. Relationship to Other Two-Species Chemotaxis and Competition Models

The quasilinear two-species model is distinct from previous classical models in several respects:

• In contrast to earlier studies where diffusion is linear, these quasilinear models yield a broader range of critical phenomena; the critical exponents generalize the well-known $v$3 mass threshold of the classical Keller–Segel system.
• Systems with flux-limited chemotaxis shift the blow-up threshold to $v$4, highlighting the effect of chemical-saturation mechanisms in suppressing singularity formation (Zeng et al., 8 Jan 2026).
• For systems with Lotka–Volterra or competition/reaction terms, global well-posedness and pattern formation mechanisms may be preserved even for large chemotactic coefficients due to the repulsive or logistic terms (Li et al., 2021, Hu et al., 2014, Issa et al., 2017).

A schematic overview of criticality for representative quasilinear two-species models is shown below:

System Class	Blow-up Criterion	Reference
Power-law diffusion, power-law sensitivity	$v$5 (for $v$6)	(Zeng et al., 8 Jan 2026)
Flux-limited chemotaxis	$v$7	(Zeng et al., 8 Jan 2026)
Power-law diffusion, linear sensitivity	$v$8	(Cai et al., 8 Jan 2026)
Two-species, linear drift with logistic	No blow-up for $v$9 arbitrary; pattern formation	(Li et al., 2021)

These results indicate sensitivity of aggregation versus global regularity to both the precise form of nonlinearity and the structure of chemotactic and competition interactions.

6. Extensions, Open Problems, and Related Research Directions

The analysis of quasilinear two-species chemotaxis models opens several avenues:

• Kinetic origin: Systematic derivation from two-population kinetic models establishes the foundational macroscopic equations and links $z$0 to microscopic motility parameters (Almeida et al., 2014).
• Pattern formation and bifurcation: Detailed study in $z$1D and $z$2D with cross-diffusion, nonlocal or competitive terms reveals Turing-type, spike, and labyrinthine aggregation patterns, with bifurcation theory and weakly nonlinear analysis elucidating pattern selection (Hu et al., 2014, Li et al., 2021).
• Traveling waves and synchronization: Analysis of coupled pulse propagation, synchronization thresholds, and invasion fronts is ongoing for quasilinear and kinetic models in one- and higher-dimensional domains (Emako et al., 2016).
• Robustness to model perturbations: Active research addresses the effect of alternative coupling (e.g., repellent responses, multi-chemical or higher-order systems), stochasticity, and boundary effects on critical exponents and solution behavior.

A central challenge remains the comprehensive classification of blow-up versus global regularity for even broader classes of nonlinearities and interaction topologies, as well as the precise characterization of singularity profiles and dynamics near critical thresholds.