TY - JOUR
T1 - Cauchy problem of a system of parabolic conservation laws arising from the singular Keller-Segel model in multi-dimensions
AU - Wang, Dehua
AU - Wang, Zhian
AU - Zhao, Kun
N1 - Funding Information:
The authors of this paper would like to thank the anonymous referee for valuable comments and suggestions, which helped improve the quality of the paper. The first author was partially supported by the National Science Foundation (grant nos. DMS-1312800 and DMS-1613213). The second author was supported by the Hong Kong RGC GRF (grant no. PolyU 153031/17P). The third author was partially supported by the Louisiana Board of Regents Research Competitiveness Subprogram (grant no. LEQSF(2015-18)-RD-A-24) and the Simons Foundation Collaboration Grant for Mathematicians (no. 413028). The third author also gratefully acknowledges start-up funding from the Department of Mathematics at Tulane University.
Publisher Copyright:
© 2021 Department of Mathematics, Indiana University. All rights reserved.
PY - 2021/1
Y1 - 2021/1
N2 - In this paper, we study the qualitative behavior of solutions to the Cauchy problem of a system of parabolic conservation laws, derived from a Keller-Segel type chemotaxis model with singular sensitivity, in multiple space dimensions. Assuming H2 initial data, it is shown that under the assumption that only some fractions of the total energy associated with the initial perturbation around a prescribed constant ground state are small, the Cauchy problem admits a unique global-in-time solution, and the solution converges to the prescribed ground state as time goes to infinity. In addition, it is shown that solutions of the fully dissipative model converge to that of the corresponding partially dissipative model with certain convergence rates as a specific system parameter tends to zero.
AB - In this paper, we study the qualitative behavior of solutions to the Cauchy problem of a system of parabolic conservation laws, derived from a Keller-Segel type chemotaxis model with singular sensitivity, in multiple space dimensions. Assuming H2 initial data, it is shown that under the assumption that only some fractions of the total energy associated with the initial perturbation around a prescribed constant ground state are small, the Cauchy problem admits a unique global-in-time solution, and the solution converges to the prescribed ground state as time goes to infinity. In addition, it is shown that solutions of the fully dissipative model converge to that of the corresponding partially dissipative model with certain convergence rates as a specific system parameter tends to zero.
KW - Cauchy problem
KW - Diffusion limit
KW - Global well-posedness
KW - Keller-Segel chemotaxis model
KW - Long-time behavior
KW - Parabolic conservation laws
UR - https://www.scopus.com/pages/publications/85102262045
U2 - 10.1512/iumj.2021.70.8075
DO - 10.1512/iumj.2021.70.8075
M3 - Journal article
AN - SCOPUS:85102262045
SN - 0022-2518
VL - 70
SP - 1
EP - 47
JO - Indiana University Mathematics Journal
JF - Indiana University Mathematics Journal
IS - 1
ER -