$W^{m,p(t,x)}$-Estimate for a Class of Higher-Order Parabolic Equations with Partially BMO Coefficients
Year: 2024
Author: Hong Tian, Shuai Hao, Shenzhou Zheng
Journal of Partial Differential Equations, Vol. 37 (2024), Iss. 2 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v37.n2.6
Journal of Partial Differential Equations, Vol. 37 (2024), Iss. 2 : pp. 198–234
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 37
Keywords: A higher-order parabolic equation Sobolev spaces with variable exponents partially BMO quasi-norm Reifenberg flat domains log-Hölder continuity.