Analysis of a parabolic cross-diffusion population model without self-diffusion

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:224

Article

Chen, L ; Jüngel, A

关键词: population equations;    strong cross-diffusion;    weak competition;    relative entropy;    global-in-time existence of weak solutions;    long-time behavior of solutions;   

DOI  :  10.1016/j.jde.2005.08.002

来源: Elsevier

【 摘 要 】

The global existence of non-negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown. The cross-diffusion terms are allowed to be arbitrarily large, whereas the self-diffusion terms are assumed to disappear. The last assumption complicates the analysis since these terms usually provide H-1 estimates of the solutions. The existence proof is based on a positivity-preserving backward Euler-Galerkin approximation, discrete entropy estimates, and L-1 weak compactness arguments. Furthermore, employing the entropy-entropy production method, we show for special stationary solutions that the transient solution converges exponentially fast to its steady state. As a by-product, we prove that only constant steady states exist if the inter-specific competition parameters disappear no matter how strong the cross-diffusion constants are. (c) 2005 Elsevier Inc. All rights reserved.

【 授权许可】

Free