{"title":"Sobolev space weak solutions to one kind of quasilinear parabolic partial differential equations related to forward-backward stochastic differential equations","authors":"Zhen Wu, Bing Xie, Zhiyong Yu","doi":"10.3934/dcds.2023018","DOIUrl":"https://doi.org/10.3934/dcds.2023018","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"8 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74683332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Center manifold theory for the 1-dimensional collective motions of camphor disks with delta functions in the $ L^2 $-framework","authors":"K. Ikeda","doi":"10.3934/dcds.2023024","DOIUrl":"https://doi.org/10.3934/dcds.2023024","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"26 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73584696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with the boundary layer problem on a chemotaxis system modelling boundary layer formation of aerobic bacteria in fluid. Completing the system with physical Robin-type boundary conditions for oxygen and no-flux boundary conditions for bacteria, we show that the gradients of its radial solutions in a region between two concentric spheres possessing boundary layer effects as the oxygen diffusion rate $ varepsilon $ goes to zero and the boundary-layer thickness is of order $ mathcal{O}(varepsilon^alpha) $ with $ 0
{"title":"Boundary layer problem on the chemotaxis model with Robin boundary conditions","authors":"Qianqian Hou","doi":"10.3934/dcds.2023108","DOIUrl":"https://doi.org/10.3934/dcds.2023108","url":null,"abstract":"This paper is concerned with the boundary layer problem on a chemotaxis system modelling boundary layer formation of aerobic bacteria in fluid. Completing the system with physical Robin-type boundary conditions for oxygen and no-flux boundary conditions for bacteria, we show that the gradients of its radial solutions in a region between two concentric spheres possessing boundary layer effects as the oxygen diffusion rate $ varepsilon $ goes to zero and the boundary-layer thickness is of order $ mathcal{O}(varepsilon^alpha) $ with $ 0<alpha<frac{1}{2} $.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"249 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135700134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem$ begin{equation*} left{ begin{array}{l} -Delta_{p(|x|)} u - Delta_{q} u = lambda (|u|^{p(|x|)-2}u + |u|^{q-2}u) quad mbox{in} mathcal B , u = 0 quad mbox{on} partial mathcal B end{array} right. end{equation*} $is considered, where $ mathcal B = { x in mathbb{R}^N : |x|1 $ on $ [0, R] $, $ Delta_{q} u = mbox{div}(|nabla u|^{q-2}nabla u) $, and $ q>1 $. The existence of positive solutions is proved for every $ lambda>lambda_1(q) $, where $ lambda_1(q) $ is the first eigenvalue of $ q $-Laplacian.
考虑$ begin{equation*} left{ begin{array}{l} -Delta_{p(|x|)} u - Delta_{q} u = lambda (|u|^{p(|x|)-2}u + |u|^{q-2}u) quad mbox{in} mathcal B , u = 0 quad mbox{on} partial mathcal B end{array} right. end{equation*} $问题，其中$ mathcal B = { x in mathbb{R}^N : |x|<R } $、$ N ge 1 $、$ Delta_{p(|x|)} u = mbox{div} (|nabla u|^{p(|x|)-2}nabla u) $、$ p(r) $是连续的，满足$ [0, R] $、$ Delta_{q} u = mbox{div}(|nabla u|^{q-2}nabla u) $、$ q>1 $上的$ p(r)>1 $。证明了每个$ lambda>lambda_1(q) $正解的存在性，其中$ lambda_1(q) $是$ q $ -拉普拉斯算子的第一个特征值。
{"title":"An eigenvalue problem for a variable exponent problem, via topological degree","authors":"Raúl Manásevich, Satoshi Tanaka","doi":"10.3934/dcds.2023134","DOIUrl":"https://doi.org/10.3934/dcds.2023134","url":null,"abstract":"The problem$ begin{equation*} left{ begin{array}{l} -Delta_{p(|x|)} u - Delta_{q} u = lambda (|u|^{p(|x|)-2}u + |u|^{q-2}u) quad mbox{in} mathcal B , u = 0 quad mbox{on} partial mathcal B end{array} right. end{equation*} $is considered, where $ mathcal B = { x in mathbb{R}^N : |x|<R } $, $ N ge 1 $, $ Delta_{p(|x|)} u = mbox{div} (|nabla u|^{p(|x|)-2}nabla u) $, $ p(r) $ is continuous and satisfies $ p(r)>1 $ on $ [0, R] $, $ Delta_{q} u = mbox{div}(|nabla u|^{q-2}nabla u) $, and $ q>1 $. The existence of positive solutions is proved for every $ lambda>lambda_1(q) $, where $ lambda_1(q) $ is the first eigenvalue of $ q $-Laplacian.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135560507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Global dynamics of evolution systems with asymptotic annihilation","authors":"Taishan Yi, Xiao-Qiang Zhao","doi":"10.3934/dcds.2023025","DOIUrl":"https://doi.org/10.3934/dcds.2023025","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"46 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77252085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Emergence of large densities in a chemotaxis system with signaling loops, nonlinear signal productions and competitions sources under nonradial symmetry case","authors":"Guangyu Xu","doi":"10.3934/dcds.2023090","DOIUrl":"https://doi.org/10.3934/dcds.2023090","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"131 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73736858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Normalized solutions for Kirchhoff type equations with combined nonlinearities: The Sobolev critical case","authors":"Xiaojing Feng, Haidong Liu, Zhitao Zhang","doi":"10.3934/dcds.2023035","DOIUrl":"https://doi.org/10.3934/dcds.2023035","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"11 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84313208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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