Existence and regularity of source-type self-similar solutions for stable thin-film equations

Interfaces and free boundaries

article

Mohamed Majdoub¹  Slim Tayachi² 

[1] Imam Abdulrahman Bin Faisal University;Université de Tunis El Manar

关键词: Fourth-order degenerate parabolic equations;    stable thin-film equations;    free boundary problems;    self-similar solutions;    source-type solutions;    existence;    regularity;   

DOI  :  10.4171/ifb/479

学科分类：生物化学工程

来源: European Mathematical Society

【 摘 要 】

We investigate the existence and the boundary regularity of source-type self-similar solutions to the thin-film equation ht=−(hnhzzz)z+(hn+3)zz,h_t=-(h^nh_{zzz})_z+(h^{n+3})_{zz},ht​=−(hnhzzz​)z​+(hn+3)zz​, t0,  z∈R;  h(0,z)=ωδ(z)0,\; z\in \mathbb{R};\; h(0,z)= \omega \delta(z)t0,z∈R;h(0,z)=ωδ(z) where n∈(3/2,3),  ω0 0n∈(3/2,3),ω0 and δ\deltaδ is the Dirac mass at the origin. It is known that the leading order expansion near the edge of the support coincides with that of a traveling-wave solution for the standard thin-film equation: ht=−(hnhzzz)zh_t=-(h^nh_{zzz})_zht​=−(hnhzzz​)z​. In this paper we sharpen this result, proving that the higher-order corrections are analytic with respect to three variables: the first one is just the {spatial} variable, whereas the second and the third (except for n=2n = 2n=2) are irrational powers of it. It is known that this third variable does not appear for the thin-film equation without gravity.

【 授权许可】

CC BY