An elliptic cross-diffusion system describing two-species models on a bounded domain with different natural conditions

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• 作者：Qi-Jian Tan ^{tanqjxxx@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Elliptic systems ; Cross-diffusion ; Discontinuous coefficients ; Reaction&ndash ; diffusion equations ; Lotka&ndash ; Volterra models ; Fixed point theorem
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 May 2016
• 年：2016
• 卷：437
• 期：2
• 页码：853-869
• 全文大小：413 K

文摘

This paper is concerned with an elliptic cross-diffusion system describing two-species models on a bounded domain Ω, where Ω consists of a finite number of subdomains ta-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000664&_mathId=si1.gif&_user=111111111&_pii=S0022247X16000664&_rdoc=1&_issn=0022247X&md5=206beb35088f05ac8b04a2a1e4b48081" title="Click to view the MathML source">Ω_itainer hidden">${\mathrm{\Omega }}_i$ (ta-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000664&_mathId=si2.gif&_user=111111111&_pii=S0022247X16000664&_rdoc=1&_issn=0022247X&md5=bfd777b2b56588490c817f4b24e49f75" title="Click to view the MathML source">i=1,…,mtainer hidden">$i=1,\dots ,m$) separated by interfaces ta-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000664&_mathId=si3.gif&_user=111111111&_pii=S0022247X16000664&_rdoc=1&_issn=0022247X&md5=5580ec7b0721134b355cc94347c67911" title="Click to view the MathML source">Γ_jtainer hidden">${\mathrm{\Gamma }}_j$ (ta-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000664&_mathId=si4.gif&_user=111111111&_pii=S0022247X16000664&_rdoc=1&_issn=0022247X&md5=6908ed2364bb247ec24d49ae527eb10c" title="Click to view the MathML source">j=1,…,m−1tainer hidden">$j=1,\dots ,m-1$) and the natural conditions of the subdomains ta-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000664&_mathId=si1.gif&_user=111111111&_pii=S0022247X16000664&_rdoc=1&_issn=0022247X&md5=206beb35088f05ac8b04a2a1e4b48081" title="Click to view the MathML source">Ω_itainer hidden">${\mathrm{\Omega }}_i$ are different. This system is strongly coupled and the coefficients of the equations are allowed to be discontinuous on interfaces ta-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000664&_mathId=si3.gif&_user=111111111&_pii=S0022247X16000664&_rdoc=1&_issn=0022247X&md5=5580ec7b0721134b355cc94347c67911" title="Click to view the MathML source">Γ_jtainer hidden">${\mathrm{\Gamma }}_j$. The main goal is to show the existence of nonnegative solutions for the system by Schauder's fixed point theorem. Furthermore, as applications, the existence of positive solutions for some Lotka–Volterra models with cross-diffusion, self-diffusion and discontinuous coefficients are also investigated.