BOUNDEDNESS OF SOLUTIONS TO A FULLY PARABOLIC KELLER-SEGEL SYSTEM WITH NONLINEAR SENSITIVITY

This paper deals with the global boundedness of solutions to a fully parabolic Keller-Segel system u(t) = Delta u-del(u(alpha)del v), v(t) = Delta v-v+u under non-flux boundary conditions in a smooth bounded domain Omega subset of R-n. The case of alpha >= max{1,2/n} with n >= 1 was considered in a previous paper of the authors [Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. B, 21 (2016), 1317-1327]. In the present paper we prove for the other case alpha epsilon (2/3,1) that if parallel to u(0)parallel to(Ln alpha/2(Omega)) and parallel to del v(0)parallel to(Ln alpha(Omega)) are small enough with n >= 3, then the solutions are globally bounded with both u and v decaying to the same constant steady (u) over bar (0) = 1/|Omega| integral u(0)(x)dx exponentially in the L-infinity-norm as t -> infinity. Moreover, the above conclusions still hold for all alpha >= 2 and n >= 1, provided parallel to u(0)parallel to(Ln alpha-n(Omega)) and parallel to del v(0)parallel to(L infinity(Omega)) sufficiently small.