Similarity Solutions for the Flux limited Keller Segel System with Time Varying Chemical Decay Rate

Technical Summary: Similarity Solutions for the Flux-limited Keller–Segel System with Time-Varying Chemical Decay Rate

Problem Statement
The classical Keller–Segel (KS) model, while foundational for chemotaxis, suffers from two critical deficiencies: the potential for finite-time blow-up of cell densities under supercritical mass conditions and the assumption of unbounded chemotactic velocity ($u = \chi \nabla v$), which violates biomechanical constraints. Flux-limited Keller–Segel (FLKS) models address these issues by imposing a maximal chemotactic speed, ensuring finite propagation and physically meaningful dynamics. However, existing FLKS analyses typically assume constant chemical decay rates. In biological reality, signal degradation is often mediated by enzymes (e.g., phosphodiesterases in Dictyostelium discoideum) whose expression levels vary over developmental cycles, leading to time-dependent decay rates $\kappa(t)$. This paper investigates a one-dimensional FLKS system where the chemical decay rate is an arbitrary function of time, aiming to determine which temporal decay patterns admit analytical similarity reductions.

Methodology
The authors employ a systematic Lie symmetry analysis combined with equivalence transformation theory.

1. Extended System Formulation: The arbitrary decay function $\kappa(t)$ is treated as an additional dependent variable subject to constraints ($\kappa_x = \kappa_u = \kappa_v = 0$), extending the phase space to $(x, t, u, v, \kappa)$.
2. Group Classification: The authors derive the determining equations for the infinitesimal generators of point transformations that map the class of FLKS equations into itself. This involves solving a system of partial differential equations for the generator coefficients and the classifying function $\kappa(t)$.
3. Optimal System Construction: For each identified case of $\kappa(t)$, the authors construct an optimal system of one-dimensional subalgebras using the adjoint representation method. This ensures a non-redundant classification of all possible similarity reductions.
4. Symmetry Reductions: The admitted symmetries are used to reduce the original system of partial differential equations (PDEs) to ordinary differential equations (ODEs) or integro-differential equations.
5. Explicit Solutions: Specific flux-limiting functions (hyperbolic tangent and Weber-Fechner forms) are utilized to derive exact or semi-analytical solutions (quadratures) for the reduced systems.

Key Contributions and Results

1. Complete Group Classification:
   The study identifies that for an arbitrary decay function $\kappa(t)$, the system admits only the kernel symmetry of spatial translation ($X_1 = \partial_x$). However, three distinguished functional forms of $\kappa(t)$ extend the symmetry algebra:

   • Constant Decay ($\kappa(t) = \kappa_0$): Admits time translation ($X_2 = \partial_t$) and spatial translation.
   • Inverse-Time (Power-Law) Decay ($\kappa(t) = \mu/t$): Admits a scaling symmetry ($X_3 = t\partial_t + \frac{1}{2}x\partial_x + \dots$), enabling self-similar reductions.
   • Exponential Decay ($\kappa(t) = \kappa_0 e^{\lambda t}$): Admits a combined time translation and chemical rescaling symmetry ($X_4 = \partial_t - \lambda v \partial_v$).

2. Derivation of Similarity Reductions:

   • Spatial Homogeneity: For all cases, spatially homogeneous solutions ($u=U(t), v=V(t)$) are derived, reducing the system to coupled ODEs solvable via integrating factors.
   • Traveling Waves: For constant and exponential decay, traveling wave solutions ($u=U(y), v=V(y)$ with $y=t-\alpha x$) are constructed. For the constant decay case with a hyperbolic tangent flux limiter, explicit closed-form solutions for cell density and chemoattractant concentration are obtained.
   • Self-Similar Solutions: For the power-law decay case, the system reduces to a coupled system of ODEs involving a bounded tanh-squared limiter. The authors derive an exact quadrature representation for the cell density, though the resulting system for the gradient involves non-local (integro-differential) terms.

3. Explicit Analytical Solutions:
   The paper provides explicit formulas for:

   • Constant Decay: Traveling wave profiles where cell density saturates due to flux limitation, balancing production and constant degradation.
   • Exponential Decay: Solutions involving the exponential integral function ($Ei$), demonstrating how up-regulation ($\lambda > 0$) can extinguish chemoattractant fields, while down-regulation ($\lambda < 0$) leads to long-lived signals.
   • Power-Law Decay: A formal representation of self-similar profiles, highlighting the transition between transient traveling waves and long-time diffusive behavior.

Significance and Claims
The authors claim that their work provides a rigorous mathematical foundation for understanding which temporal decay patterns in chemotactic signaling admit similarity reductions. The study demonstrates that:

• Time-dependent decay is not merely a mathematical regularization but encodes biologically meaningful control strategies (e.g., enzymatic regulation).
• Specific decay laws (constant, power-law, exponential) correspond to distinct dynamical regimes: bounded traveling fronts, self-similar redistribution, and rapid suppression or amplification of cues, respectively.
• The symmetry-based approach enables analytical progress on flux-limited models with realistic, time-varying degradation mechanisms, offering benchmark solutions for validating numerical schemes in higher dimensions.

Limitations and Future Directions
The paper modestly notes that the self-similar reduction for the power-law decay case leads to an integro-differential system that does not admit a purely local ODE reduction under generic flux-limiting functions. Consequently, the explicit solution formulas for this case are interpreted as formal representations requiring additional closure assumptions or numerical integration. The authors identify a complete analysis of this specific case as an open direction for future investigation.