|This dissertation is devoted to the study of the classical Keller-Segel chemotaxis systems with space-time heterogeneous logistic source function on $R^N$. Chemotaxis systems are mathematical models describing aggregation phenomena of cells due to chemotaxis. That is, phenomena of directed movement of cells in response to the gradient of a chemical attractant, which may be produced by the cells themselves.
We first study the fundamental problems such as local existence and global existence of nonnegative classical solutions for given nonnegative initial function in various spaces. Among our results, we prove that it is enough for the self-limitation coefficient of the logistic source function to be greater than or equal to the chemotaxis sensitivity coefficient to guarantee the existence of time-global classical solutions.
Next, we discuss the pointwise and uniform persistence of classical solutions, the existence of positive entire solutions, the existence of time-periodic solution if the logistic function is time-periodic, and, the existence of steady state solutions if the logistic function is time homogeneous. In particular, we show that any classical solution with a positive initial function enjoys pointwise persistence under the same assumption of the existence of time-global classical solution. Moreover, we study the stability of positive entire solutions, and the spreading feature of solutions with compactly supported or front like initials. In this direction, our results recover as a special case the stability and spreading speeds for the classical Fisher-KPP equations.
Finally, we establish the existence and non-existence of traveling wave solutions. When the logistic function is homogeneous and the chemotaxis sensitivity coefficient is sufficiently small, we show that there are traveling wave solutions with arbitrarily large speeds and there is no traveling wave solution of arbitrarily small speeds. That is there are positive constant $0<c^*_-\leq c^*_+<\infty$ such that for any $c\geq c^*_+$, there is a traveling wave solution with speed $c$ connecting the two trivial constant solutions and no such solutions exist with speed $c<c^*_-$.