Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on $\mathbb{R}^{N}$

Abstract

The current paper is concerned with the spreading speeds of the following parabolicparabolic chemotaxis model with logistic source on R N ,where χ, a, b, λ, µ are positive constants. Assume b > N µχ 4 . Among others, it is proved that 2 √ a is the spreading speed of the global classical solutions of (0.1) with nonempty compactly supported initial functions, that is,where (u(x, t; u 0 , v 0 ), v(x, t; u 0 , v 0 )) is the unique global classical solution of (0.1) with u(x, 0; u 0 , v 0 ) = u 0 , v(x, 0; u 0 , v 0 ) = v 0 , and supp(u 0 ), supp(v 0 ) are nonempty and compact. It is well known that 2 √ a is the spreading speed of the following Fisher-KPP equation,Hence, if b > N µχ 4 , the chemotaxis neither speeds up nor slows down the spatial spreading in the Fisher-KPP equation.