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 - <p style="text-align: justify;">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.</p>