Time Asymptotic Behavior of Solutions to a Chemotaxis Model with Logarithmic Singularity

The paper is a continuation of the author’s work in [14]. We consider a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity, non-diffusive chemical signal and density-dependent production/consumption rate. We consider Cauchy problems with Cauchy data not bounded away from the logarithmic singularity. The model can be converted into a \(2\times 2\) system of hyperbolic-parabolic balance laws by inverse Hopf-Cole transformation, with Cauchy data connecting two different end states. The converted form was studied in [14] when Cauchy data are near a diffusive contact wave. The current paper is to study the scenario under the original model to gain understanding of the evolution of physical quantities when the logarithmic singularity plays an intrinsic role. For all three cases, singularity at \(-\infty \) , at \(+\infty \) , and at \(\pm \infty \) , we obtain a clear picture of time asymptotic behavior of solutions.