Parabolic-Elliptic Chemotaxis Models with Singular Sensitivity

Abstract

Chemotaxis describes the movement of biological cells or organisms in response to certain chemicals in their environments and occupies an essential role in coordinating cell movement in many biological circumstances such as tumor growth, immune system response, embryo development, population dynamics, gravitational collapse. Chemotaxis models, also known as Keller-Segel models, have been widely studied since the pioneering works by Keller and Segel at the beginning of 1970s on the mathematical modeling of the aggregation process of Dictyostelium discoideum.

One of the central problems on chemotaxis models is whether solutions blow up in finite time or exist globally. The second essential question is whether solutions are bounded if they exist globally. Moreover, if that is true, what is the asymptotic behavior of globally defined bounded positive solutions over time. In recent years, a large amount of research has been carried out toward those problems in various chemotaxis models. However, many interesting problems still remain open associated to those central questions. As far as we know, there has been no study on the parabolic-elliptic chemotaxis models with singular sensitivity and kinetics term in any dimensional setting. This dissertation aims to study the dynamics of one species chemotaxis model with singular sensitivity in bounded heterogeneous environments, and two species chemotaxis model with singular sensitivity in bounded homogeneous environments to provide answers to those central questions in different scenarios.

Regarding chemotaxis model with singular sensitivity of one species in heterogeneous environments, we first study global existence of classical solutions. Among our results, we prove that in any space dimensional setting, logistic kinetics is sufficient to enforce the global existence of classical solutions without requiring any condition on the parameters and hence prevents the occurrence of finite-time blow-up. Next, we discuss the boundedness of classical solutions, the mass and uniform persistence of classical solutions, and the existence of positive entire solutions. In particular, under some explicit assumption on the parameters, we show that any globally defined positive solution is bounded above and below eventually by some positive constants which are independent of its initial functions. Finally, we prove the existence of positive entire solutions under the same condition on the parameters.

Regarding chemotaxis model with singular sensitivity for two competing species in homogeneous environments, we first investigate global existence of classical solutions and prove that in general dimensional setting, solutions globally exist with no restrictions on the parameters as in the one species case. Next, under some explicit assumption on the parameters, we establish uniform boundedness, combined mass and uniform persistence of classical solutions, which implies that there are some positive bounds independent of its initial function from above and below for the sum of solutions.

To establish the qualitative properties of Keller–Segel type chemotaxis models mentioned in the above, we have developed several novel techniques and strategies, which enable us to obtain outstanding results that go beyond what has previously been achieved for both one species and two competition species chemotaxis models.