具有驱动延迟项的非自治反应扩散方程的回拉吸引子

IF 2.1 3区数学 Q1 MATHEMATICS, APPLIED

Mathematical Methods in the Applied Sciences Pub Date : 2025-02-25 DOI:10.1002/mma.10843

Yong Ren, Yongqin Xie, Jiangwei Zhang

{"title":"具有驱动延迟项的非自治反应扩散方程的回拉吸引子","authors":"Yong Ren, Yongqin Xie, Jiangwei Zhang","doi":"10.1002/mma.10843","DOIUrl":null,"url":null,"abstract":"<div>\n \n In this paper, we mainly investigate the asymptotic behavior of nonautonomous reaction–diffusion equation with the driving delay term in whole space. A new method (or technique) is introduced for verifying the \n<math>\n <semantics>\n <mrow>\n <mfenced>\n <mrow>\n <msub>\n <mrow>\n <mi>C</mi>\n </mrow>\n <mrow>\n <msup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n </msup>\n <mo>(</mo>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mi>N</mi>\n </mrow>\n </msup>\n <mo>)</mo>\n </mrow>\n </msub>\n <mo>,</mo>\n <msub>\n <mrow>\n <mi>C</mi>\n </mrow>\n <mrow>\n <msup>\n <mrow>\n <mi>H</mi>\n </mrow>\n <mrow>\n <mn>1</mn>\n </mrow>\n </msup>\n <mo>(</mo>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mi>N</mi>\n </mrow>\n </msup>\n <mo>)</mo>\n </mrow>\n </msub>\n </mrow>\n </mfenced>\n </mrow>\n <annotation>$$ \\left({C}_{L&amp;amp;#x0005E;2\\left({\\mathbb{R}}&amp;amp;#x0005E;N\\right)},{C}_{H&amp;amp;#x0005E;1\\left({\\mathbb{R}}&amp;amp;#x0005E;N\\right)}\\right) $$</annotation>\n </semantics></math>-pullback \n<math>\n <mrow>\n <mi>𝒟</mi>\n </mrow></math>-asymptotic compactness of the family of processes (see Theorem 2). As an application, the \n<math>\n <semantics>\n <mrow>\n <mfenced>\n <mrow>\n <msub>\n <mrow>\n <mi>C</mi>\n </mrow>\n <mrow>\n <msup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n </msup>\n <mo>(</mo>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mi>N</mi>\n </mrow>\n </msup>\n <mo>)</mo>\n </mrow>\n </msub>\n <mo>,</mo>\n <msub>\n <mrow>\n <mi>C</mi>\n </mrow>\n <mrow>\n <msup>\n <mrow>\n <mi>H</mi>\n </mrow>\n <mrow>\n <mn>1</mn>\n </mrow>\n </msup>\n <mo>(</mo>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mi>N</mi>\n </mrow>\n </msup>\n <mo>)</mo>\n </mrow>\n </msub>\n </mrow>\n </mfenced>\n </mrow>\n <annotation>$$ \\left({C}_{L&amp;amp;#x0005E;2\\left({\\mathbb{R}}&amp;amp;#x0005E;N\\right)},{C}_{H&amp;amp;#x0005E;1\\left({\\mathbb{R}}&amp;amp;#x0005E;N\\right)}\\right) $$</annotation>\n </semantics></math>-pullback \n<math>\n <mrow>\n <mi>𝒟</mi>\n </mrow></math>-attractor is obtained. In particular, the nonlinearity \n<math>\n <semantics>\n <mrow>\n <mi>f</mi>\n </mrow>\n <annotation>$$ f $$</annotation>\n </semantics></math> satisfies the polynomial growth of arbitrary order, and the delay term \n<math>\n <semantics>\n <mrow>\n <mi>g</mi>\n <mo>(</mo>\n <mi>t</mi>\n <mo>,</mo>\n <msub>\n <mrow>\n <mi>u</mi>\n </mrow>\n <mrow>\n <mi>t</mi>\n </mrow>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$$ g\\left(t,{u}_t\\right) $$</annotation>\n </semantics></math> may be driven by a function with very weak assumptions, namely, just measurability, which deepens some previous results.\n </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 9","pages":"9796-9808"},"PeriodicalIF":2.1000,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Pullback Attractors for Nonautonomous Reaction–Diffusion Equations With the Driving Delay Term in ℝN\",\"authors\":\"Yong Ren, Yongqin Xie, Jiangwei Zhang\",\"doi\":\"10.1002/mma.10843\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>\\n \\n In this paper, we mainly investigate the asymptotic behavior of nonautonomous reaction–diffusion equation with the driving delay term in whole space. A new method (or technique) is introduced for verifying the \\n<math>\\n <semantics>\\n <mrow>\\n <mfenced>\\n <mrow>\\n <msub>\\n <mrow>\\n <mi>C</mi>\\n </mrow>\\n <mrow>\\n <msup>\\n <mrow>\\n <mi>L</mi>\\n </mrow>\\n <mrow>\\n <mn>2</mn>\\n </mrow>\\n </msup>\\n <mo>(</mo>\\n <msup>\\n <mrow>\\n <mi>ℝ</mi>\\n </mrow>\\n <mrow>\\n <mi>N</mi>\\n </mrow>\\n </msup>\\n <mo>)</mo>\\n </mrow>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mrow>\\n <mi>C</mi>\\n </mrow>\\n <mrow>\\n <msup>\\n <mrow>\\n <mi>H</mi>\\n </mrow>\\n <mrow>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n <mo>(</mo>\\n <msup>\\n <mrow>\\n <mi>ℝ</mi>\\n </mrow>\\n <mrow>\\n <mi>N</mi>\\n </mrow>\\n </msup>\\n <mo>)</mo>\\n </mrow>\\n </msub>\\n </mrow>\\n </mfenced>\\n </mrow>\\n <annotation>$$ \\\\left({C}_{L&amp;amp;#x0005E;2\\\\left({\\\\mathbb{R}}&amp;amp;#x0005E;N\\\\right)},{C}_{H&amp;amp;#x0005E;1\\\\left({\\\\mathbb{R}}&amp;amp;#x0005E;N\\\\right)}\\\\right) $$</annotation>\\n </semantics></math>-pullback \\n<math>\\n <mrow>\\n <mi>𝒟</mi>\\n </mrow></math>-asymptotic compactness of the family of processes (see Theorem 2). As an application, the \\n<math>\\n <semantics>\\n <mrow>\\n <mfenced>\\n <mrow>\\n <msub>\\n <mrow>\\n <mi>C</mi>\\n </mrow>\\n <mrow>\\n <msup>\\n <mrow>\\n <mi>L</mi>\\n </mrow>\\n <mrow>\\n <mn>2</mn>\\n </mrow>\\n </msup>\\n <mo>(</mo>\\n <msup>\\n <mrow>\\n <mi>ℝ</mi>\\n </mrow>\\n <mrow>\\n <mi>N</mi>\\n </mrow>\\n </msup>\\n <mo>)</mo>\\n </mrow>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mrow>\\n <mi>C</mi>\\n </mrow>\\n <mrow>\\n <msup>\\n <mrow>\\n <mi>H</mi>\\n </mrow>\\n <mrow>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n <mo>(</mo>\\n <msup>\\n <mrow>\\n <mi>ℝ</mi>\\n </mrow>\\n <mrow>\\n <mi>N</mi>\\n </mrow>\\n </msup>\\n <mo>)</mo>\\n </mrow>\\n </msub>\\n </mrow>\\n </mfenced>\\n </mrow>\\n <annotation>$$ \\\\left({C}_{L&amp;amp;#x0005E;2\\\\left({\\\\mathbb{R}}&amp;amp;#x0005E;N\\\\right)},{C}_{H&amp;amp;#x0005E;1\\\\left({\\\\mathbb{R}}&amp;amp;#x0005E;N\\\\right)}\\\\right) $$</annotation>\\n </semantics></math>-pullback \\n<math>\\n <mrow>\\n <mi>𝒟</mi>\\n </mrow></math>-attractor is obtained. In particular, the nonlinearity \\n<math>\\n <semantics>\\n <mrow>\\n <mi>f</mi>\\n </mrow>\\n <annotation>$$ f $$</annotation>\\n </semantics></math> satisfies the polynomial growth of arbitrary order, and the delay term \\n<math>\\n <semantics>\\n <mrow>\\n <mi>g</mi>\\n <mo>(</mo>\\n <mi>t</mi>\\n <mo>,</mo>\\n <msub>\\n <mrow>\\n <mi>u</mi>\\n </mrow>\\n <mrow>\\n <mi>t</mi>\\n </mrow>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$$ g\\\\left(t,{u}_t\\\\right) $$</annotation>\\n </semantics></math> may be driven by a function with very weak assumptions, namely, just measurability, which deepens some previous results.\\n </div>\",\"PeriodicalId\":49865,\"journal\":{\"name\":\"Mathematical Methods in the Applied Sciences\",\"volume\":\"48 9\",\"pages\":\"9796-9808\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2025-02-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Methods in the Applied Sciences\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/mma.10843\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Methods in the Applied Sciences","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mma.10843","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

摘要

本文主要研究具有驱动时滞项的非自治反应扩散方程在全空间上的渐近行为。介绍了一种新的验证cl2的方法（或技术）(n)，C h 1 ((N) $$ \left({C}_{L&amp;#x0005E;2\left({\mathbb{R}}&amp;#x0005E;N\right)},{C}_{H&amp;#x0005E;1\left({\mathbb{R}}&amp;#x0005E;N\right)}\right) $$ -回拉性-过程族的渐近紧性（见定理2）。作为一个应用程序，lc2(n)，C h 1 (得到$$ \left({C}_{L&amp;#x0005E;2\left({\mathbb{R}}&amp;#x0005E;N\right)},{C}_{H&amp;#x0005E;1\left({\mathbb{R}}&amp;#x0005E;N\right)}\right) $$ -pull - back -attractor。其中，非线性f $$ f $$满足任意阶多项式增长，延时项g (t)，但t) $$ g\left(t,{u}_t\right) $$可能是由一个假设非常弱的函数驱动的，即，只是可测量性，这加深了以前的一些结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Pullback Attractors for Nonautonomous Reaction–Diffusion Equations With the Driving Delay Term in ℝN

In this paper, we mainly investigate the asymptotic behavior of nonautonomous reaction–diffusion equation with the driving delay term in whole space. A new method (or technique) is introduced for verifying the $(C_{L^{2} (ℝ^{N})}, C_{H^{1} (ℝ^{N})})$ -pullback $𝒟$ -asymptotic compactness of the family of processes (see Theorem 2). As an application, the $(C_{L^{2} (ℝ^{N})}, C_{H^{1} (ℝ^{N})})$ -pullback $𝒟$ -attractor is obtained. In particular, the nonlinearity $f$ satisfies the polynomial growth of arbitrary order, and the delay term $g (t, u_{t})$ may be driven by a function with very weak assumptions, namely, just measurability, which deepens some previous results.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Mathematical Methods in the Applied Sciences 数学-应用数学

CiteScore

4.90

自引率

6.90%

发文量

798

审稿时长

6 months

期刊介绍： Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.