Convergence to diffusion waves for solutions of 1D KellerSegel model
Abstract
In this paper, we are concerned with the asymptotic behavior of solutions to the Cauchy problem (or initialboundary value problem) of onedimensional KellerSegel model. For the Cauchy problem, we prove that the solutions timeasymptotically converge to the nonlinear diffusion wave whose profile is selfsimilar solution to the corresponding parabolic equation, which is derived by Darcy's law, as in [11, 28]. For the initialboundary value problem, we consider two cases: Dirichlet boundary condition and null Neumann boundary condition on (u, \rho). In the case of Dirichlet boundary condition, similar to the Cauchy problem, the asymptotic profile is still the selfsimilar solution of the corresponding parabolic equation, which is derived by Darcy's law, thus we only need to deal with boundary effect. In the case of nullNeumann boundary condition, the global existence and asymptotic behavior of solutions near constant steady states are established. The proof is based on the elementary energy method and some delicate analysis of the corresponding asymptotic profiles.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.11317
 Bibcode:
 2021arXiv210911317L
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics
 EPrint:
 33 pages