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Volume 37, Issue 2
$W^{m,p(t,x)}$-Estimate for a Class of Higher-Order Parabolic Equations with Partially BMO Coefficients

Hong Tian, Shuai Hao & Shenzhou Zheng

DOI: 10.4208/jpde.v37.n2.6

J. Part. Diff. Eq., 37 (2024), pp. 198-234.

Published online: 2024-06

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  • Abstract

We prove a global estimate in the Sobolev spaces with variable exponents to the solution of a class of higher-order divergence parabolic equations with measurable coefficients over the non-smooth domains. Here, it is mainly assumed that the coefficients are allowed to be merely measurable in one of the spatial variables and have a small BMO quasi-norm in the other variables at a sufficiently small scale, while the boundary of the underlying domain belongs to the so-called Reifenberg flatness. This is a natural outgrowth of Dong-Kim-Zhang’s papers [1, 2] from the $W^{m,p}$-regularity to the $W^{m,p(t,x)}$-regularity for such higher-order parabolic equations with merely measurable coefficients with Reifenberg flat domain which is beyond the Lipschitz domain with small Lipschitz constant.

  • Keywords

A higher-order parabolic equation, Sobolev spaces with variable exponents, partially BMO quasi-norm, Reifenberg flat domains, log-Hölder continuity.

  • AMS Subject Headings

35B65, 35K25, 35R05, 46E30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-37-198, author = {Tian , HongHao , Shuai and Zheng , Shenzhou}, title = {$W^{m,p(t,x)}$-Estimate for a Class of Higher-Order Parabolic Equations with Partially BMO Coefficients}, journal = {Journal of Partial Differential Equations}, year = {2024}, volume = {37}, number = {2}, pages = {198--234}, abstract = {

We prove a global estimate in the Sobolev spaces with variable exponents to the solution of a class of higher-order divergence parabolic equations with measurable coefficients over the non-smooth domains. Here, it is mainly assumed that the coefficients are allowed to be merely measurable in one of the spatial variables and have a small BMO quasi-norm in the other variables at a sufficiently small scale, while the boundary of the underlying domain belongs to the so-called Reifenberg flatness. This is a natural outgrowth of Dong-Kim-Zhang’s papers [1, 2] from the $W^{m,p}$-regularity to the $W^{m,p(t,x)}$-regularity for such higher-order parabolic equations with merely measurable coefficients with Reifenberg flat domain which is beyond the Lipschitz domain with small Lipschitz constant.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v37.n2.6}, url = {http://global-sci.org/intro/article_detail/jpde/23209.html} }
TY - JOUR T1 - $W^{m,p(t,x)}$-Estimate for a Class of Higher-Order Parabolic Equations with Partially BMO Coefficients AU - Tian , Hong AU - Hao , Shuai AU - Zheng , Shenzhou JO - Journal of Partial Differential Equations VL - 2 SP - 198 EP - 234 PY - 2024 DA - 2024/06 SN - 37 DO - http://doi.org/10.4208/jpde.v37.n2.6 UR - https://global-sci.org/intro/article_detail/jpde/23209.html KW - A higher-order parabolic equation, Sobolev spaces with variable exponents, partially BMO quasi-norm, Reifenberg flat domains, log-Hölder continuity. AB -

We prove a global estimate in the Sobolev spaces with variable exponents to the solution of a class of higher-order divergence parabolic equations with measurable coefficients over the non-smooth domains. Here, it is mainly assumed that the coefficients are allowed to be merely measurable in one of the spatial variables and have a small BMO quasi-norm in the other variables at a sufficiently small scale, while the boundary of the underlying domain belongs to the so-called Reifenberg flatness. This is a natural outgrowth of Dong-Kim-Zhang’s papers [1, 2] from the $W^{m,p}$-regularity to the $W^{m,p(t,x)}$-regularity for such higher-order parabolic equations with merely measurable coefficients with Reifenberg flat domain which is beyond the Lipschitz domain with small Lipschitz constant.

Tian , HongHao , Shuai and Zheng , Shenzhou. (2024). $W^{m,p(t,x)}$-Estimate for a Class of Higher-Order Parabolic Equations with Partially BMO Coefficients. Journal of Partial Differential Equations. 37 (2). 198-234. doi:10.4208/jpde.v37.n2.6
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