On the logarithmic Keller-Segel-Fisher/KPP system

Abstract

1.1. Background. System (1.1) is derived from the following Keller-Segel type chemotaxis model with logarithmic sensitivity and logistic growth:where the unknown functions and system parameters are interpreted as follows: • u(x, t): density of cellular population at space location x and time t, • c(x, t): concentration of chemical signal at space location x and time t, • D > 0: diffusion coefficient of cellular population, • χ = 0: coefficient of chemotactic sensitivity, • a > 0: relative growth rate of cellular population, • K > 0: relative carrying capacity of cellular population, • ε ≥ 0: diffusion coefficient of chemical signal, • µ = 0: coefficient of density-dependent production/degradation rate of chemical signal, • σ ≥ 0: natural degradation rate of chemical signal.

Recent results for the logarithmic Keller-Segel-Fisher/KPP System

“…Our contribution in this regard is some recent studies on the Keller-Segel-Fisher/KPP model (1.10) (hence (1.9)) concerning the existence of global-in-time solution, long-time behavior, vanishing coefficient limit and optimal time decay rates of the solution, without or relaxing the smallness assumption on the initial perturbation [15,16]. The results are to be detailed in next section.…”

“…Theorem 2.2 (Global Existence [15]). Consider the Cauchy problem (2.1), where r, D > 0, χ = 0 and ε ≥ 0 are fixed constants.…”

“…Theorem 2.4 (Zero Diffusion Limit and Convergence Rate [15]). Let the conditions of Theorem 2.2 hold.…”

“…Theorem 2.5 (Decay Rates [15]). Let the conditions of Theorem 2.2 hold, and assume that ψ 0 ≡ ψ(·, 0) ∈ L 2 (R).…”

“…Since there is no difference in the analysis of these two scenarios, we assume χµ > 0 throughout this paper. Mathematically, the non-diffusive part of the transformed system to be discussed below is hyperbolic in biologically relevant regimes when χµ > 0, while it may change type when χµ < 0 [15]. For Othmer-Stevens' model (1.2), the logarithmic function can be removed by the inverse Hopf-Cole transformation [2]:…”