Geodesic
Existence and regularity of source-type self-similar solutions for stable thin-film equations
Mohamed Majdoub1 ; Slim Tayachi2
1Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia
2Université de Tunis El Manar, Tunisia
Interfaces and free boundaries, Tome 24 (2022) no. 3, pp. 431-457

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DOI

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​, t>0,z∈R;h(0,z)=ωδ(z) where n∈(3/2,3),ω>0 and δ 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​)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=2) are irrational powers of it. It is known that this third variable does not appear for the thin-film equation without gravity.
DOI : 10.4171/ifb/479
Classification : 35-XX, 34-XX
Mots-clés : Fourth-order degenerate parabolic equations, stable thin-film equations, free boundary problems, self-similar solutions, source-type solutions, existence, regularity

Mohamed Majdoub  1   ; Slim Tayachi  2

1 Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia
2 Université de Tunis El Manar, Tunisia 
Mohamed Majdoub; Slim Tayachi. Existence and regularity of source-type self-similar solutions for stable thin-film equations. Interfaces and free boundaries, Tome 24 (2022) no. 3, pp. 431-457. doi: 10.4171/ifb/479
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     title = {Existence and regularity of source-type self-similar solutions for stable thin-film equations},
     journal = {Interfaces and free boundaries},
     pages = {431--457},
     year = {2022},
     volume = {24},
     number = {3},
     doi = {10.4171/ifb/479},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/479/}
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