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

摘要

This paper deals with the global boundedness of solutions to a fully parabolic Keller-Segel system u_t = ∆u - ▽(u~α▽_v), v_t = ∆v - v + u under non-flux boundary conditions in a smooth bounded domain Ω ⊂ R~n. The case of α ≥ max{1, 2} 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 α ϵ (2/3,1) that if ||u_0||_(L~((nα))/2(Ω) and ||▽v_0||_(L~(nα)(Ω) are small enough with n ≥ 3, then the solutions are globally bounded with both u and v decaying to the same constant steady state u_0 = 1/Ω ∫_Ωu_0(x)dx exponentially in the L~∞-norm as t → ∞. Moreover, the above conclusions still hold for all α ≥ 2 and n ≥ 1, provided ||u_0||_(L~(nα-n)(Ω) and ||▽v_0||_L~∞(Ω) sufficiently small.