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{"title":"具有非线性扩散和奇异敏感性的二维趋化系统中的全局有界性","authors":"Guoqiang Ren, Xing Zhou","doi":"10.1515/anona-2023-0125","DOIUrl":null,"url":null,"abstract":"\n <jats:p>In this study, we investigate the two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity: <jats:disp-formula id=\"j_anona-2023-0125_eq_001\">\n <jats:alternatives>\n <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0125_eq_001.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\n <m:mfenced open=\"{\" close=\"\">\n <m:mrow>\n <m:mtable displaystyle=\"true\">\n <m:mtr>\n <m:mtd columnalign=\"left\">\n <m:msub>\n <m:mrow>\n <m:mi>u</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>t</m:mi>\n </m:mrow>\n </m:msub>\n <m:mo>=</m:mo>\n <m:mrow>\n <m:mo>∇</m:mo>\n </m:mrow>\n <m:mo>⋅</m:mo>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:msup>\n <m:mrow>\n <m:mi>u</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>θ</m:mi>\n <m:mo>−</m:mo>\n <m:mn>1</m:mn>\n </m:mrow>\n </m:msup>\n <m:mrow>\n <m:mo>∇</m:mo>\n </m:mrow>\n <m:mi>u</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:mo>−</m:mo>\n <m:mi>χ</m:mi>\n <m:mrow>\n <m:mo>∇</m:mo>\n </m:mrow>\n <m:mo>⋅</m:mo>\n <m:mfenced open=\"(\" close=\")\">\n <m:mrow>\n <m:mfrac>\n <m:mrow>\n <m:mi>u</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>v</m:mi>\n </m:mrow>\n </m:mfrac>\n <m:mrow>\n <m:mo>∇</m:mo>\n </m:mrow>\n <m:mi>v</m:mi>\n </m:mrow>\n </m:mfenced>\n <m:mo>,</m:mo>\n </m:mtd>\n <m:mtd columnalign=\"left\">\n <m:mi>x</m:mi>\n <m:mo>∈</m:mo>\n <m:mi mathvariant=\"normal\">Ω</m:mi>\n <m:mo>,</m:mo>\n <m:mspace width=\"0.33em\" />\n <m:mi>t</m:mi>\n <m:mo>></m:mo>\n <m:mn>0</m:mn>\n <m:mo>,</m:mo>\n </m:mtd>\n </m:mtr>\n <m:mtr>\n <m:mtd columnalign=\"left\">\n <m:msub>\n <m:mrow>\n <m:mi>v</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>t</m:mi>\n </m:mrow>\n </m:msub>\n <m:mo>=</m:mo>\n <m:mi mathvariant=\"normal\">Δ</m:mi>\n <m:mi>v</m:mi>\n <m:mo>−</m:mo>\n <m:mi>v</m:mi>\n <m:mo>+</m:mo>\n <m:mi>u</m:mi>\n <m:mo>+</m:mo>\n <m:mi>g</m:mi>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi>x</m:mi>\n <m:mo>,</m:mo>\n <m:mi>t</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:mo>,</m:mo>\n </m:mtd>\n <m:mtd columnalign=\"left\">\n <m:mi>x</m:mi>\n <m:mo>∈</m:mo>\n <m:mi mathvariant=\"normal\">Ω</m:mi>\n <m:mo>,</m:mo>\n <m:mspace width=\"0.33em\" />\n <m:mi>t</m:mi>\n <m:mo>></m:mo>\n <m:mn>0</m:mn>\n <m:mo>,</m:mo>\n </m:mtd>\n </m:mtr>\n <m:mtr>\n <m:mtd columnalign=\"left\" />\n </m:mtr>\n </m:mtable>\n </m:mrow>\n </m:mfenced>\n <m:mspace width=\"2.0em\" />\n <m:mspace width=\"2.0em\" />\n <m:mspace width=\"2.0em\" />\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mo>∗</m:mo>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n </m:math>\n <jats:tex-math>\\left\\{\\begin{array}{ll}{u}_{t}=\\nabla \\cdot \\left({u}^{\\theta -1}\\nabla u)-\\chi \\nabla \\cdot \\left(\\frac{u}{v}\\nabla v\\right),& x\\in \\Omega ,\\hspace{0.33em}t\\gt 0,\\\\ {v}_{t}=\\Delta v-v+u+g\\left(x,t),& x\\in \\Omega ,\\hspace{0.33em}t\\gt 0,\\\\ \\end{array}\\right.\\hspace{2.0em}\\hspace{2.0em}\\hspace{2.0em}\\left(\\ast )</jats:tex-math>\n </jats:alternatives>\n </jats:disp-formula> in a bounded domain with smooth boundary. We present the global boundedness of weak solutions to the model (<jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0125_eq_002.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mo>∗</m:mo>\n </m:math>\n <jats:tex-math>\\ast </jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>) if <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0125_eq_003.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>θ</m:mi>\n <m:mo>></m:mo>\n <m:mfrac>\n <m:mrow>\n <m:mn>3</m:mn>\n </m:mrow>\n <m:mrow>\n <m:mn>2</m:mn>\n </m:mrow>\n </m:mfrac>\n </m:math>\n <jats:tex-math>\\theta \\gt \\frac{3}{2}</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> and (1.10)–(1.11). This result improves our recent work.</jats:p>","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":3.2000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity\",\"authors\":\"Guoqiang Ren, Xing Zhou\",\"doi\":\"10.1515/anona-2023-0125\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"\\n <jats:p>In this study, we investigate the two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity: <jats:disp-formula id=\\\"j_anona-2023-0125_eq_001\\\">\\n <jats:alternatives>\\n <jats:graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0125_eq_001.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"block\\\">\\n <m:mfenced open=\\\"{\\\" close=\\\"\\\">\\n <m:mrow>\\n <m:mtable displaystyle=\\\"true\\\">\\n <m:mtr>\\n <m:mtd columnalign=\\\"left\\\">\\n <m:msub>\\n <m:mrow>\\n <m:mi>u</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mi>t</m:mi>\\n </m:mrow>\\n </m:msub>\\n <m:mo>=</m:mo>\\n <m:mrow>\\n <m:mo>∇</m:mo>\\n </m:mrow>\\n <m:mo>⋅</m:mo>\\n <m:mrow>\\n <m:mo>(</m:mo>\\n <m:mrow>\\n <m:msup>\\n <m:mrow>\\n <m:mi>u</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mi>θ</m:mi>\\n <m:mo>−</m:mo>\\n <m:mn>1</m:mn>\\n </m:mrow>\\n </m:msup>\\n <m:mrow>\\n <m:mo>∇</m:mo>\\n </m:mrow>\\n <m:mi>u</m:mi>\\n </m:mrow>\\n <m:mo>)</m:mo>\\n </m:mrow>\\n <m:mo>−</m:mo>\\n <m:mi>χ</m:mi>\\n <m:mrow>\\n <m:mo>∇</m:mo>\\n </m:mrow>\\n <m:mo>⋅</m:mo>\\n <m:mfenced open=\\\"(\\\" close=\\\")\\\">\\n <m:mrow>\\n <m:mfrac>\\n <m:mrow>\\n <m:mi>u</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mi>v</m:mi>\\n </m:mrow>\\n </m:mfrac>\\n <m:mrow>\\n <m:mo>∇</m:mo>\\n </m:mrow>\\n <m:mi>v</m:mi>\\n </m:mrow>\\n </m:mfenced>\\n <m:mo>,</m:mo>\\n </m:mtd>\\n <m:mtd columnalign=\\\"left\\\">\\n <m:mi>x</m:mi>\\n <m:mo>∈</m:mo>\\n <m:mi mathvariant=\\\"normal\\\">Ω</m:mi>\\n <m:mo>,</m:mo>\\n <m:mspace width=\\\"0.33em\\\" />\\n <m:mi>t</m:mi>\\n <m:mo>></m:mo>\\n <m:mn>0</m:mn>\\n <m:mo>,</m:mo>\\n </m:mtd>\\n </m:mtr>\\n <m:mtr>\\n <m:mtd columnalign=\\\"left\\\">\\n <m:msub>\\n <m:mrow>\\n <m:mi>v</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mi>t</m:mi>\\n </m:mrow>\\n </m:msub>\\n <m:mo>=</m:mo>\\n <m:mi mathvariant=\\\"normal\\\">Δ</m:mi>\\n <m:mi>v</m:mi>\\n <m:mo>−</m:mo>\\n <m:mi>v</m:mi>\\n <m:mo>+</m:mo>\\n <m:mi>u</m:mi>\\n <m:mo>+</m:mo>\\n <m:mi>g</m:mi>\\n <m:mrow>\\n <m:mo>(</m:mo>\\n <m:mrow>\\n <m:mi>x</m:mi>\\n <m:mo>,</m:mo>\\n <m:mi>t</m:mi>\\n </m:mrow>\\n <m:mo>)</m:mo>\\n </m:mrow>\\n <m:mo>,</m:mo>\\n </m:mtd>\\n <m:mtd columnalign=\\\"left\\\">\\n <m:mi>x</m:mi>\\n <m:mo>∈</m:mo>\\n <m:mi mathvariant=\\\"normal\\\">Ω</m:mi>\\n <m:mo>,</m:mo>\\n <m:mspace width=\\\"0.33em\\\" />\\n <m:mi>t</m:mi>\\n <m:mo>></m:mo>\\n <m:mn>0</m:mn>\\n <m:mo>,</m:mo>\\n </m:mtd>\\n </m:mtr>\\n <m:mtr>\\n <m:mtd columnalign=\\\"left\\\" />\\n </m:mtr>\\n </m:mtable>\\n </m:mrow>\\n </m:mfenced>\\n <m:mspace width=\\\"2.0em\\\" />\\n <m:mspace width=\\\"2.0em\\\" />\\n <m:mspace width=\\\"2.0em\\\" />\\n <m:mrow>\\n <m:mo>(</m:mo>\\n <m:mrow>\\n <m:mo>∗</m:mo>\\n </m:mrow>\\n <m:mo>)</m:mo>\\n </m:mrow>\\n </m:math>\\n <jats:tex-math>\\\\left\\\\{\\\\begin{array}{ll}{u}_{t}=\\\\nabla \\\\cdot \\\\left({u}^{\\\\theta -1}\\\\nabla u)-\\\\chi \\\\nabla \\\\cdot \\\\left(\\\\frac{u}{v}\\\\nabla v\\\\right),& x\\\\in \\\\Omega ,\\\\hspace{0.33em}t\\\\gt 0,\\\\\\\\ {v}_{t}=\\\\Delta v-v+u+g\\\\left(x,t),& x\\\\in \\\\Omega ,\\\\hspace{0.33em}t\\\\gt 0,\\\\\\\\ \\\\end{array}\\\\right.\\\\hspace{2.0em}\\\\hspace{2.0em}\\\\hspace{2.0em}\\\\left(\\\\ast )</jats:tex-math>\\n </jats:alternatives>\\n </jats:disp-formula> in a bounded domain with smooth boundary. We present the global boundedness of weak solutions to the model (<jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0125_eq_002.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mo>∗</m:mo>\\n </m:math>\\n <jats:tex-math>\\\\ast </jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula>) if <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0125_eq_003.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>θ</m:mi>\\n <m:mo>></m:mo>\\n <m:mfrac>\\n <m:mrow>\\n <m:mn>3</m:mn>\\n </m:mrow>\\n <m:mrow>\\n <m:mn>2</m:mn>\\n </m:mrow>\\n </m:mfrac>\\n </m:math>\\n <jats:tex-math>\\\\theta \\\\gt \\\\frac{3}{2}</jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula> and (1.10)–(1.11). This result improves our recent work.</jats:p>\",\"PeriodicalId\":51301,\"journal\":{\"name\":\"Advances in Nonlinear Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.2000,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Nonlinear Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/anona-2023-0125\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Nonlinear Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/anona-2023-0125","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
In this study, we investigate the two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity:
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\left\{\begin{array}{ll}{u}_{t}=\nabla \cdot \left({u}^{\theta -1}\nabla u)-\chi \nabla \cdot \left(\frac{u}{v}\nabla v\right),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {v}_{t}=\Delta v-v+u+g\left(x,t),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ \end{array}\right.\hspace{2.0em}\hspace{2.0em}\hspace{2.0em}\left(\ast )
in a bounded domain with smooth boundary. We present the global boundedness of weak solutions to the model (
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and (1.10)–(1.11). This result improves our recent work.