A parabolic–elliptic chemotaxis system with non-negative chemotactic sensitivity χ∕v and positive chemical diffusion coefficient η is considered in a smooth bounded domain Ω⊂R N , N≥2 under homogeneous Neumann boundary conditions. We show the existence of at least one global weak solution for χ<χ N , N≥3, where χ N ≔[Formula presented]. Moreover, under further assumptions of Ω and η, we prove that the constructed solution becomes smooth and stabilizes to a constant steady state after some waiting time if N=3,4. The stabilization of a global bounded solution, and the non-existence of non-constant steady states are also discussed in general dimensions.
Bibliographical noteFunding Information:
The authors would like to thank the referees for their comments and suggestions on the improvement of the paper. JA was supported by NRF, Republic of Korea - 2018R1D1A1B07047465 . KK was supported by NRF, Republic of Korea - 2017R1A2B4006484 and NRF, Republic of Korea - 2015R1A5A1009350 . JL was supported by SSTF-BA1701-05 ( Samsung Science & Technology Foundation, Republic of Korea ).
All Science Journal Classification (ASJC) codes
- Economics, Econometrics and Finance(all)
- Computational Mathematics
- Applied Mathematics