Optimal partial boundary condition for degenerate parabolic equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:284

Article

Zhan, Huashui¹  Feng, Zhaosheng² 

[1] Xiamen Univ Technol, Sch Appl Math, Xiamen 361024, Fujian, Peoples R China

[2] Univ Texas Rio Grande Valley, Sch Math & Stat Sci, Edinburg, TX 78539 USA

关键词: Non-Newtonian fluid equation;    Stability;    Gronwall's inequality;    Partial boundary value condition;    Degenerate parabolic equation;   

DOI  :  10.1016/j.jde.2021.02.053

来源: Elsevier

【 摘 要 】

For the stability of the non-Newtonian fluid equation partial derivative u/partial derivative t - div(a(x)vertical bar del u vertical bar(p-2)del u) - Sigma(N)(i=1) b(i)(x)D(i)u + c(x, t)u = f (x, t), where a(x)vertical bar(x is an element of Omega) > 0, a(x)vertical bar x is an element of partial derivative Omega = 0 and b(i)(x) is an element of C-1((Omega) over bar), we know that the degeneracy of a(x) may make the usual Dirichlet boundary value condition overdetermined and only a partial boundary value condition is expected. How to depict the geometric characteristic of the partial boundary value condition has been a long-time standing open problem. In this study, an optimal partial boundary value condition has been proposed, and the stability of weak solutions based on this partial boundary value condition is established. When the rate of the diffusion coefficient decays to zero, we explore how it affects the stability of weak solutions. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free