Higher fractional differentiability for solutions to parabolic equations with double-phase growth

IF 1.8 3区 数学 Q1 MATHEMATICS, APPLIED Nonlinear Analysis-Real World Applications Pub Date : 2024-11-30 DOI:10.1016/j.nonrwa.2024.104270
Lijing Zhao, Shenzhou Zheng
{"title":"Higher fractional differentiability for solutions to parabolic equations with double-phase growth","authors":"Lijing Zhao,&nbsp;Shenzhou Zheng","doi":"10.1016/j.nonrwa.2024.104270","DOIUrl":null,"url":null,"abstract":"<div><div>We devote this paper to a higher fractional differentiability of solutions for a class of parabolic double-phase equations <span><span><span><math><mrow><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>−</mo><mtext>div</mtext><mfenced><mrow><msup><mrow><mrow><mo>|</mo><mi>D</mi><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>D</mi><mi>u</mi><mo>+</mo><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>D</mi><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>D</mi><mi>u</mi></mrow></mfenced><mo>=</mo><mo>−</mo><mtext>div</mtext><mfenced><mrow><msup><mrow><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>F</mi><mo>+</mo><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>F</mi></mrow></mfenced><mspace></mspace><mtext>in</mtext><mspace></mspace><mspace></mspace><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>.</mo></mrow></math></span></span></span>A higher fractional differentiability of spatial gradients is established by way of the finite difference quotient, under assumptions that <span><math><mrow><mn>0</mn><mo>≤</mo><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>α</mi><mo>,</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span>, the exponents <span><math><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></math></span> satisfies <span><math><mrow><mn>2</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>q</mi><mo>≤</mo><mi>p</mi><mo>+</mo><mfrac><mrow><mn>2</mn><mi>α</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></math></span>, and <span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> belongs to <span><math><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>l</mi><mi>o</mi><mi>c</mi></mrow><mrow><mi>ϑ</mi></mrow></msubsup><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>T</mi><mo>;</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>Φ</mi><mo>,</mo><mi>∞</mi><mo>;</mo><mi>l</mi><mi>o</mi><mi>c</mi></mrow><mrow><mspace></mspace><mi>β</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mn>0</mn><mo>&lt;</mo><mi>β</mi><mo>&lt;</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>ϑ</mi><mo>≔</mo><mo>max</mo><mrow><mo>{</mo><mfrac><mrow><mi>q</mi><mrow><mo>(</mo><mn>2</mn><mi>q</mi><mo>−</mo><mi>p</mi><mo>)</mo></mrow></mrow><mrow><mi>p</mi></mrow></mfrac><mo>,</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span>, where <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>Φ</mi><mo>,</mo><mi>∞</mi><mo>;</mo><mi>l</mi><mi>o</mi><mi>c</mi></mrow><mrow><mspace></mspace><mi>β</mi></mrow></msubsup></math></span> is the local Besov-Orlicz space.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104270"},"PeriodicalIF":1.8000,"publicationDate":"2024-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Analysis-Real World Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1468121824002098","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

Abstract

We devote this paper to a higher fractional differentiability of solutions for a class of parabolic double-phase equations tudiv|Du|p2Du+a(x,t)|Du|q2Du=div|F|p2F+a(x,t)|F|q2FinΩT.A higher fractional differentiability of spatial gradients is established by way of the finite difference quotient, under assumptions that 0a(x,t)Cα,α2(ΩT) for α(0,1], the exponents p,q satisfies 2pqp+2αn+2, and F(x,t) belongs to Llocϑ(0,T;BΦ,;locβ(Ω)) for 0<β<1 and ϑmax{q(2qp)p,q+1}, where BΦ,;locβ is the local Besov-Orlicz space.
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
双相增长抛物型方程解的高分数可微性
本文研究一类抛物型双相方程∂tu - div|Du|p - 2Du+a(x,t)|Du|q - 2Du= - div|F|p - 2F+a(x,t)|F|q−2FinΩT解的高分数可微性。在假设0≤A (x,t)∈Cα,α2(ΩT)对于α∈(0,1),指数p,q满足2≤p≤q≤p+2αn+2, F(x,t)属于lloc(0, t; BΦ,∞;loloc(Ω))对于0<;β<1,和{q(2q−p)p,q+1},其中BΦ,∞;loloc是局部besovo - orlicz空间,利用有限差分商建立空间梯度的高分数可微性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
CiteScore
3.80
自引率
5.00%
发文量
176
审稿时长
59 days
期刊介绍: Nonlinear Analysis: Real World Applications welcomes all research articles of the highest quality with special emphasis on applying techniques of nonlinear analysis to model and to treat nonlinear phenomena with which nature confronts us. Coverage of applications includes any branch of science and technology such as solid and fluid mechanics, material science, mathematical biology and chemistry, control theory, and inverse problems. The aim of Nonlinear Analysis: Real World Applications is to publish articles which are predominantly devoted to employing methods and techniques from analysis, including partial differential equations, functional analysis, dynamical systems and evolution equations, calculus of variations, and bifurcations theory.
期刊最新文献
Enhanced dissipation and temporal decay in the Euler–Poisson–Navier–Stokes equations Optimality of smallness conditions in Willmore obstacle problems under Dirichlet boundary conditions A Hyperbolic–parabolic framework to manage traffic generated pollution Existence of weak solutions for nonisothermal immiscible compressible two-phase flow in porous media On the onset of wave-breaking and the time evolution of the maximum of horizontal velocity in rotational equatorial waves
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1