{"title":"Existence of traveling wave solutions for density-dependent diffusion competitive systems","authors":"Yang Wang, Xuanyu Lv, Fan Liu, Xiaoguang Zhang","doi":"10.1088/1361-6544/ad6acd","DOIUrl":null,"url":null,"abstract":"In this paper we are concerned with the existence of traveling wave solutions for two species competitive systems with density-dependent diffusion. Since the density-dependent diffusion is a kind of nonlinear diffusion and degenerates at the origin, the methods for proving the existence of traveling wave solutions for competitive systems with linear diffusion are invalid. To overcome the degeneracy of diffusion, we construct a nonlinear invariant region Ω near the origin. Then by using the method of phase plane analysis, we prove the existence of traveling wave solutions connecting the origin and the unique coexistence state, when the speed <italic toggle=\"yes\">c</italic> is large than some positive value. In addition, when one species is density-dependent diffusive while the other is linear diffusive, via the change of variables and the central manifold theorem, we prove the existence of the minimal speed <inline-formula>\n<tex-math><?CDATA $c^*$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"nonad6acdieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>. And for <inline-formula>\n<tex-math><?CDATA $c\\unicode{x2A7E} c^*$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mi>c</mml:mi><mml:mtext>⩾</mml:mtext><mml:msup><mml:mi>c</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"nonad6acdieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>, traveling wave solutions connecting the origin and the unique coexistence state still exist. In particular, when <inline-formula>\n<tex-math><?CDATA $c = c^*$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>c</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"nonad6acdieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>, we find that one component of the traveling wave solution is sharp type while the other component is smooth, which is a different phenomenon from linear diffusive systems and scalar equations.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"15 1","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinearity","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1088/1361-6544/ad6acd","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we are concerned with the existence of traveling wave solutions for two species competitive systems with density-dependent diffusion. Since the density-dependent diffusion is a kind of nonlinear diffusion and degenerates at the origin, the methods for proving the existence of traveling wave solutions for competitive systems with linear diffusion are invalid. To overcome the degeneracy of diffusion, we construct a nonlinear invariant region Ω near the origin. Then by using the method of phase plane analysis, we prove the existence of traveling wave solutions connecting the origin and the unique coexistence state, when the speed c is large than some positive value. In addition, when one species is density-dependent diffusive while the other is linear diffusive, via the change of variables and the central manifold theorem, we prove the existence of the minimal speed c∗. And for c⩾c∗, traveling wave solutions connecting the origin and the unique coexistence state still exist. In particular, when c=c∗, we find that one component of the traveling wave solution is sharp type while the other component is smooth, which is a different phenomenon from linear diffusive systems and scalar equations.
期刊介绍:
Aimed primarily at mathematicians and physicists interested in research on nonlinear phenomena, the journal''s coverage ranges from proofs of important theorems to papers presenting ideas, conjectures and numerical or physical experiments of significant physical and mathematical interest.
Subject coverage:
The journal publishes papers on nonlinear mathematics, mathematical physics, experimental physics, theoretical physics and other areas in the sciences where nonlinear phenomena are of fundamental importance. A more detailed indication is given by the subject interests of the Editorial Board members, which are listed in every issue of the journal.
Due to the broad scope of Nonlinearity, and in order to make all papers published in the journal accessible to its wide readership, authors are required to provide sufficient introductory material in their paper. This material should contain enough detail and background information to place their research into context and to make it understandable to scientists working on nonlinear phenomena.
Nonlinearity is a journal of the Institute of Physics and the London Mathematical Society.