论 Musielak-Orlicz 空间中的一类双非线性演化方程

IF 0.8 3区 数学 Q2 MATHEMATICS
Goro Akagi, Giulio Schimperna
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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,&nbsp;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. 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引用次数: 0

摘要

本文涉及一个抛物线演化方程,其形式为 A(ut)+B(u)=f$A(u_t) + B(u) = f$,在 Rd$\mathbb {R}^d$ 的光滑有界域中求解,d≥1$d\ge 1$,并辅以初始条件和(为简单起见)同相 Dirichlet 边界条件。这里,-B$-B$ 代表扩散算子,可能是非线性的,其范围很广,包括拉普拉茨算子、适合 m∈(1,∞)$m\in (1,\infty)$ 的 m$m$-Laplacian 算子、"可变分量 "m(x)$m(x)$-Laplacian 算子,甚至一些分数阶算子。假定算子 A$A$ 的形式为 [A(v)](x,t)=α(x,v(x,t))$[A(v)](x,t)=\alpha (x,v(x,t))$ α$\alpha$在 x$x$ 中是可测的,在 v$v$ 中是最大单调的。主要结果致力于证明一大类函数 α$\alpha$ 的弱解的存在性,扩展了之前与变指数情况相关的结果所考虑的环境,即 α(x,v)=|v(x)|p(x)-2v(x)$\alpha (x,v)=|v(x)|^{p(x)-2}v(x)$ 。为此,我们将在满足所谓 Δ2$\Delta _2$型结构条件的穆西拉克-奥利兹空间中建立亚微分算子理论,并建立一个近似作用于该类空间的最大单调算子的框架。然后,应用这种理论为特定方程提供一个存在性结果,但它本身可能具有独立的意义。最后,我们将提出一些具体方程(以及相应的算子 A$A$、B$B$)来说明存在性结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On a class of doubly nonlinear evolution equations in Musielak–Orlicz spaces

This paper is concerned with a parabolic evolution equation of the form A ( u t ) + B ( u ) = f $A(u_t) + B(u) = f$ , settled in a smooth bounded domain of R d $\mathbb {R}^d$ , d 1 $d\ge 1$ , and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, B $-B$ stands for a diffusion operator, possibly nonlinear, which may range in a very wide class, including the Laplacian, the m $m$ -Laplacian for suitable m ( 1 , ) $m\in (1,\infty)$ , the “variable-exponent” m ( x ) $m(x)$ -Laplacian, or even some fractional order operators. The operator A $A$ is assumed to be in the form [ A ( v ) ] ( x , t ) = α ( x , v ( x , t ) ) $[A(v)](x,t)=\alpha (x,v(x,t))$ with α $\alpha$ being measurable in x $x$ and maximal monotone in v $v$ . The main results are devoted to proving existence of weak solutions for a wide class of functions α $\alpha$ that extends the setting considered in previous results related to the variable exponent case where α ( x , v ) = | v ( x ) | p ( x ) 2 v ( x ) $\alpha (x,v)=|v(x)|^{p(x)-2}v(x)$ . To this end, a theory of subdifferential operators will be established in Musielak–Orlicz spaces satisfying structure conditions of the so-called Δ 2 $\Delta _2$ -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 A $A$ , B $B$ ) to which the result can be applied.

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来源期刊
CiteScore
1.50
自引率
0.00%
发文量
157
审稿时长
4-8 weeks
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
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