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Register a new userExistence and regularity of source-type self--similar solutions for the thin-film equation with gravity
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We investigate the existence and the boundary regularity of source-type self-similar solutions to the thin-film equation $h_t=-(h^nh_{zzz})_z+(h^{n+3})_{zz},$ $ t>0,, zin R;, h(0,z)= M delta$ where $nin (3/2,3),, M > 0$ and $delta$ 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 travelling-wave solution for the standard thin-film equation: $h_t=-(h^nh_{zzz})_z$. 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 spacial variable, whereas the second and third (except for $n = 2$) are irrational powers of it. It is known that this third order term does not appear for the thin-film equation without gravity.
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