关于一个分数阶薄膜方程
On a fractional thin film equation
摘要：
本文研究了一类阶为$α$的非线性退化抛物型方程，它是薄膜方程的一种分数阶形式。实际上，这一个对应于极限值$α=4$，而多孔介质方程是极限值$α=2$。本文证明了在一般初值条件下非负弱解的存在性，并建立了它的主要性质。我们还构造了自相似形式的特解，这些特解是显式的，并且是紧支撑的。在多孔介质的情况下，他们应该给长时间的行为或广泛的解决方案。最后一个结果在某些假设下被证明是正确的。最后，我们考虑具有相同非线性结构但阶数为4到6的非局部方程。对于这些方程，我们构造的自相似的解决方案，是积极的和compensately支持，从而有助于高阶理论。
英文摘要：
This paper deals with a nonlinear degenerate parabolic equation of order $α$ between 2 and 4 which is a kind of fractional version of the Thin Film Equation. Actually, this one corresponds to the limit value $α=4$ while the Porous Medium Equation is the limit $α=2$. We prove existence of a nonnegative weak solution for a general class of initial data, and establish its main properties. We also construct the special solutions in self-similar form which turn out to be explicit and compactly supported. As in the porous medium case, they are supposed to give the long time behaviour or the wide class of solutions. This last result is proved to be true under some assumptions. Lastly, we consider nonlocal equations with the same nonlinear structure but with order from 4 to 6. For these equations we construct self-similar solutions that are positive and compactly supported, thus contributing to the higher order theory.
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