A repertoire of repulsive Keller–Segel models with logarithmic sensitivity: Derivation, traveling waves, and quasi-stationary dynamics

Abstract

In this paper, we show how the chemotactic model

{partial derivative(t)rho = d(1) Delta(x)rho - del(x) . (rho del(x)c)

partial derivative(t)c = d(2) Delta(x)c + F(rho, c, del(x)rho, del(x)c, Delta x rho)

introduced in Alejo and Lopez (2021), which accounts for a chemical production-degradation operator of Hamilton-Jacobi type involving first- and second-order derivatives of the logarithm of the cell concentration, namely,

F = mu + tau c - sigma rho + A Delta(x)rho/rho + B vertical bar del(x)rho vertical bar(2)/rho(2) + C vertical bar del(x)c vertical bar(2),

with mu, tau, sigma, A, B, C is an element of R, can be formally reduced to a repulsive Keller-Segel model with logarithmic sensitivity

{ partial derivative(t)rho = D-1 Delta(x)rho + chi del(x) . (rho del(x) log(c)), chi, lambda, beta > 0,

partial derivative(t)c = D-2 Delta(x)c + lambda rho c - beta c

whenever the chemotactic parameters are appropriately chosen and the cell concentration keeps strictly positive. In this way, some explicit solutions (namely, traveling waves and stationary cell density profiles) of the former system can be transferred to a number of variants of the the latter by means of an adequate change of variables.