  In this paper, we study the spatial propagation dynamics of a
parabolic-elliptic chemotaxis system with logistic source which reduces to the
well-known Fisher-KPP equation without chemotaxis. It is known that for fast
decaying initial functions, this system has a finite spreading speed. For
slowly decaying initial functions, we show that the accelerating propagation
will occur and chemotaxis does not affect the propagation mode determined by
slowly decaying initial functions if the logistic damping is strong, that is,
the system has the same upper and lower bounds of the accelerating propagation
as for the classical Fisher-KPP equation. The main new idea of proving our
results is the construction of auxiliary equations to overcome the lack of
comparison principle due to chemotaxis.
