### Abstract

An approach to some “optimal” (more precisely, non-improvable) regularity of solutions of the thin film equation
View the MathML sourceut=−∇⋅(|u|n∇Δu)inRN×R+,u(x,0)=u0(x)inRN,
where n∈(0,2)n∈(0,2) is a fixed exponent, with smooth compactly supported initial data u0(x)u0(x), in dimensions N≥2N≥2 is discussed. Namely, a precise exponent for the Hölder continuity with respect to the spatial radial variable |x||x| is obtained by construction of a Graveleau-type focusing self-similar solution. As a consequence, optimal regularity of the gradient ∇u in certain LpLp spaces, as well as a Hölder continuity property of solutions with respect to x and t , are derived, which cannot be obtained by classic standard methods of integral identities–inequalities. Several profiles for the solutions in the cases n=0n=0 and n>0n>0 are also plotted. In general, we claim that, even for arbitrarily small n>0n>0 and positive analytic initial data u0(x)u0(x), the solutions u(x,t)u(x,t) cannot be better than View the MathML sourceCx2−ε-smooth, where ε(n)=O(n)ε(n)=O(n) as n→0n→0.

Original language | English |
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Pages (from-to) | 1099-1123 |

Number of pages | 25 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 431 |

Issue number | 2 |

Early online date | 18 Jun 2015 |

DOIs | |

Publication status | Published - 15 Nov 2015 |

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## Cite this

Alvarez-Caudevilla, P., Evans, J. D., & Galaktionov, V. A. (2015). Towards optimal regularity for the fourth-order thin film equation in RN: Graveleau-type focusing self-similarity.

*Journal of Mathematical Analysis and Applications*,*431*(2), 1099-1123. https://doi.org/10.1016/j.jmaa.2015.06.027