Abstract
We consider a Keller–Segel type chemotaxis model with logarithmic sensitivity and logistic growth. The logarithmic singularity in the system is removed via the inverse Hopf–Cole transformation. We then linearize the system around a constant equilibrium state, and obtain a detailed, pointwise description of the Green’s function. The result provides a complete solution picture for the linear problem. It also helps to shed light on small solutions of the nonlinear system.
Highlights
We consider a Keller–Segel type chemotaxis model with logarithmic sensitivity and logistic growth: ct = εcxx − μuc − σc, ut + χ[u(ln c)x]x = Duxx au(1 − u K ), x ∈ R, t > 0. (1)Here, the unknown functions c = c(x, t) and u = u(x, t) are the concentration of a chemical signal and the density of a cellular population, respectively
The system parameters are interpreted as follows
When χ < 0 and μ < 0, as adopted in [1] for the non-growth model, it describes the movement of cells that deposit a chemical signal to modify the local environment for succeeding passages
Summary
When χ < 0 and μ < 0, as adopted in [1] for the non-growth model, it describes the movement of cells that deposit a chemical signal to modify the local environment for succeeding passages. Such a scenario has found applications in cancer research [2]. Where the Cauchy datum (v0, u0) is assumed to be a small perturbation of a constant equilibrium state (v, u). To study small data solutions, especially their long time behavior, one needs to study the corresponding linear system, linearized around the constant equilibrium state. The cases correspond to different types of systems: hyperbolic–parabolic conservation laws, hyperbolic balance laws, and hyperbolic–parabolic balance laws
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