We consider the initial value problem for a system of parabolic partial differential equations modeling chemotaxis in $\R^n (n\ge 1)$, and give the asymptotic profiles for a specific class of solutions by space-time higher-order asymptotic expansions.
"Higher-order asymptotic expansions for a parabolic system modeling chemotaxis in the whole space." Hiroshima Math. J. 39 (3) 363 - 420, November 2009. https://doi.org/10.32917/hmj/1257544214