{"title":"论 Musielak-Orlicz 空间中的一类双非线性演化方程","authors":"Goro Akagi, Giulio Schimperna","doi":"10.1002/mana.202300374","DOIUrl":null,"url":null,"abstract":"<p>This paper is concerned with a parabolic evolution equation of the form <span></span><math>\n <semantics>\n <mrow>\n <mi>A</mi>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>u</mi>\n <mi>t</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n <mo>+</mo>\n <mi>B</mi>\n <mrow>\n <mo>(</mo>\n <mi>u</mi>\n <mo>)</mo>\n </mrow>\n <mo>=</mo>\n <mi>f</mi>\n </mrow>\n <annotation>$A(u_t) + B(u) = f$</annotation>\n </semantics></math>, settled in a smooth bounded domain of <span></span><math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mi>d</mi>\n </msup>\n <annotation>$\\mathbb {R}^d$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>≥</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$d\\ge 1$</annotation>\n </semantics></math>, and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, <span></span><math>\n <semantics>\n <mrow>\n <mo>−</mo>\n <mi>B</mi>\n </mrow>\n <annotation>$-B$</annotation>\n </semantics></math> stands for a diffusion operator, possibly nonlinear, which may range in a very wide class, including the Laplacian, the <span></span><math>\n <semantics>\n <mi>m</mi>\n <annotation>$m$</annotation>\n </semantics></math>-Laplacian for suitable <span></span><math>\n <semantics>\n <mrow>\n <mi>m</mi>\n <mo>∈</mo>\n <mo>(</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mi>∞</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$m\\in (1,\\infty)$</annotation>\n </semantics></math>, the “variable-exponent” <span></span><math>\n <semantics>\n <mrow>\n <mi>m</mi>\n <mo>(</mo>\n <mi>x</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$m(x)$</annotation>\n </semantics></math>-Laplacian, or even some fractional order operators. The operator <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> is assumed to be in the form <span></span><math>\n <semantics>\n <mrow>\n <mo>[</mo>\n <mi>A</mi>\n <mo>(</mo>\n <mi>v</mi>\n <mo>)</mo>\n <mo>]</mo>\n <mo>(</mo>\n <mi>x</mi>\n <mo>,</mo>\n <mi>t</mi>\n <mo>)</mo>\n <mo>=</mo>\n <mi>α</mi>\n <mo>(</mo>\n <mi>x</mi>\n <mo>,</mo>\n <mi>v</mi>\n <mo>(</mo>\n <mi>x</mi>\n <mo>,</mo>\n <mi>t</mi>\n <mo>)</mo>\n <mo>)</mo>\n </mrow>\n <annotation>$[A(v)](x,t)=\\alpha (x,v(x,t))$</annotation>\n </semantics></math> with <span></span><math>\n <semantics>\n <mi>α</mi>\n <annotation>$\\alpha$</annotation>\n </semantics></math> being measurable in <span></span><math>\n <semantics>\n <mi>x</mi>\n <annotation>$x$</annotation>\n </semantics></math> and maximal monotone in <span></span><math>\n <semantics>\n <mi>v</mi>\n <annotation>$v$</annotation>\n </semantics></math>. The main results are devoted to proving existence of weak solutions for a wide class of functions <span></span><math>\n <semantics>\n <mi>α</mi>\n <annotation>$\\alpha$</annotation>\n </semantics></math> that extends the setting considered in previous results related to the variable exponent case where <span></span><math>\n <semantics>\n <mrow>\n <mi>α</mi>\n <mrow>\n <mo>(</mo>\n <mi>x</mi>\n <mo>,</mo>\n <mi>v</mi>\n <mo>)</mo>\n </mrow>\n <mo>=</mo>\n <msup>\n <mrow>\n <mo>|</mo>\n <mi>v</mi>\n <mrow>\n <mo>(</mo>\n <mi>x</mi>\n <mo>)</mo>\n </mrow>\n <mo>|</mo>\n </mrow>\n <mrow>\n <mi>p</mi>\n <mo>(</mo>\n <mi>x</mi>\n <mo>)</mo>\n <mo>−</mo>\n <mn>2</mn>\n </mrow>\n </msup>\n <mi>v</mi>\n <mrow>\n <mo>(</mo>\n <mi>x</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\alpha (x,v)=|v(x)|^{p(x)-2}v(x)$</annotation>\n </semantics></math>. To this end, a theory of subdifferential operators will be established in Musielak–Orlicz spaces satisfying structure conditions of the so-called <span></span><math>\n <semantics>\n <msub>\n <mi>Δ</mi>\n <mn>2</mn>\n </msub>\n <annotation>$\\Delta _2$</annotation>\n </semantics></math>-type, and a framework for approximating maximal monotone operators acting in that class of spaces will also be developed. Such a theory is then applied to provide an existence result for a specific equation, but it may have an independent interest in itself. Finally, the existence result is illustrated by presenting a number of specific equations (and, correspondingly, of operators <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mi>B</mi>\n <annotation>$B$</annotation>\n </semantics></math>) to which the result can be applied.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 7","pages":"2686-2729"},"PeriodicalIF":0.8000,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On a class of doubly nonlinear evolution equations in Musielak–Orlicz spaces\",\"authors\":\"Goro Akagi, Giulio Schimperna\",\"doi\":\"10.1002/mana.202300374\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This paper is concerned with a parabolic evolution equation of the form <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>A</mi>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>u</mi>\\n <mi>t</mi>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <mo>+</mo>\\n <mi>B</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>u</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>=</mo>\\n <mi>f</mi>\\n </mrow>\\n <annotation>$A(u_t) + B(u) = f$</annotation>\\n </semantics></math>, settled in a smooth bounded domain of <span></span><math>\\n <semantics>\\n <msup>\\n <mi>R</mi>\\n <mi>d</mi>\\n </msup>\\n <annotation>$\\\\mathbb {R}^d$</annotation>\\n </semantics></math>, <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n <mo>≥</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$d\\\\ge 1$</annotation>\\n </semantics></math>, and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>−</mo>\\n <mi>B</mi>\\n </mrow>\\n <annotation>$-B$</annotation>\\n </semantics></math> stands for a diffusion operator, possibly nonlinear, which may range in a very wide class, including the Laplacian, the <span></span><math>\\n <semantics>\\n <mi>m</mi>\\n <annotation>$m$</annotation>\\n </semantics></math>-Laplacian for suitable <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>m</mi>\\n <mo>∈</mo>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>,</mo>\\n <mi>∞</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$m\\\\in (1,\\\\infty)$</annotation>\\n </semantics></math>, the “variable-exponent” <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>m</mi>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$m(x)$</annotation>\\n </semantics></math>-Laplacian, or even some fractional order operators. The operator <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math> is assumed to be in the form <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>[</mo>\\n <mi>A</mi>\\n <mo>(</mo>\\n <mi>v</mi>\\n <mo>)</mo>\\n <mo>]</mo>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>,</mo>\\n <mi>t</mi>\\n <mo>)</mo>\\n <mo>=</mo>\\n <mi>α</mi>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>,</mo>\\n <mi>v</mi>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>,</mo>\\n <mi>t</mi>\\n <mo>)</mo>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$[A(v)](x,t)=\\\\alpha (x,v(x,t))$</annotation>\\n </semantics></math> with <span></span><math>\\n <semantics>\\n <mi>α</mi>\\n <annotation>$\\\\alpha$</annotation>\\n </semantics></math> being measurable in <span></span><math>\\n <semantics>\\n <mi>x</mi>\\n <annotation>$x$</annotation>\\n </semantics></math> and maximal monotone in <span></span><math>\\n <semantics>\\n <mi>v</mi>\\n <annotation>$v$</annotation>\\n </semantics></math>. The main results are devoted to proving existence of weak solutions for a wide class of functions <span></span><math>\\n <semantics>\\n <mi>α</mi>\\n <annotation>$\\\\alpha$</annotation>\\n </semantics></math> that extends the setting considered in previous results related to the variable exponent case where <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>α</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>,</mo>\\n <mi>v</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>=</mo>\\n <msup>\\n <mrow>\\n <mo>|</mo>\\n <mi>v</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>|</mo>\\n </mrow>\\n <mrow>\\n <mi>p</mi>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>)</mo>\\n <mo>−</mo>\\n <mn>2</mn>\\n </mrow>\\n </msup>\\n <mi>v</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\alpha (x,v)=|v(x)|^{p(x)-2}v(x)$</annotation>\\n </semantics></math>. To this end, a theory of subdifferential operators will be established in Musielak–Orlicz spaces satisfying structure conditions of the so-called <span></span><math>\\n <semantics>\\n <msub>\\n <mi>Δ</mi>\\n <mn>2</mn>\\n </msub>\\n <annotation>$\\\\Delta _2$</annotation>\\n </semantics></math>-type, and a framework for approximating maximal monotone operators acting in that class of spaces will also be developed. Such a theory is then applied to provide an existence result for a specific equation, but it may have an independent interest in itself. Finally, the existence result is illustrated by presenting a number of specific equations (and, correspondingly, of operators <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math>, <span></span><math>\\n <semantics>\\n <mi>B</mi>\\n <annotation>$B$</annotation>\\n </semantics></math>) to which the result can be applied.</p>\",\"PeriodicalId\":49853,\"journal\":{\"name\":\"Mathematische Nachrichten\",\"volume\":\"297 7\",\"pages\":\"2686-2729\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-03-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematische Nachrichten\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/mana.202300374\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Nachrichten","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202300374","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
On a class of doubly nonlinear evolution equations in Musielak–Orlicz spaces
This paper is concerned with a parabolic evolution equation of the form , settled in a smooth bounded domain of , , and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, stands for a diffusion operator, possibly nonlinear, which may range in a very wide class, including the Laplacian, the -Laplacian for suitable , the “variable-exponent” -Laplacian, or even some fractional order operators. The operator is assumed to be in the form with being measurable in and maximal monotone in . The main results are devoted to proving existence of weak solutions for a wide class of functions that extends the setting considered in previous results related to the variable exponent case where . To this end, a theory of subdifferential operators will be established in Musielak–Orlicz spaces satisfying structure conditions of the so-called -type, and a framework for approximating maximal monotone operators acting in that class of spaces will also be developed. Such a theory is then applied to provide an existence result for a specific equation, but it may have an independent interest in itself. Finally, the existence result is illustrated by presenting a number of specific equations (and, correspondingly, of operators , ) to which the result can be applied.
期刊介绍:
Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index