Nonlinear Degenerate Parabolic Problems and Related Topics

P. Daskalopoulos 摘要如下： 本文的主要目的是研究一类拟线性或完全非线性退化抛物方程的解， 更复杂的微分几何问题，包括里奇流 和高斯曲率流这些问题在以下方面有实际应用： 例如群体动力学、多孔介质中的扩散和薄液膜动力学。 其中一个具体的领域将被调查的是在自由边界问题所产生的退化的某些非线性抛物方程，包括多孔介质方程，发展的p-Laplacian方程和高斯曲率流与平坦的两侧的正则性问题。 主要目标的另一个具体领域涉及超快扩散抛物方程的研究，由于 他们的关系，微分几何的主题，如里奇流和Yamabe流。 这些方程的柯西问题的可解性， 研究了解的爆破剖面、非径向结构解和 解的唯一性，是将要研究的问题之一 在这个特定的领域。 符号解的唯一性问题 在多孔介质方程中， 关于一类线性奇异抛物方程的连续性问题，福尔斯属于所提出的主要研究目标。 根据这一建议所研究的非线性方程组形成了许多应用的基本概念，这些应用被认为对技术和整个社会都很重要。 从化学品到石油甚至水的材料的净化通常通过过滤器的扩散来实现。 净化过滤器为建议书中描述的多孔介质。 薄膜动力学和作用于薄膜之间的货车范德华力由奇异的准线性超快微分方程描述。 种群增长、高分子链增长，包括生物分子的交联和高速增长的动力学，也是我们基础研究的非线性现象。 膨胀宇宙和其他宇宙学现象的有趣问题似乎是由非线性动力学支配的，在某些情况下，它是这里描述的更复杂问题的应用。

DMS-9801304 P. Daskalopoulos The abstact follows : The major objectives of this proposal are concerned with the study of certain quasilinear or fully-nonlinear degenerate parabolic equations in connection with more complexproblems of differential geometry, including the Ricci flow and the Gauss curvature flow. These problems find physical applications in such areas as population dynamics, diffusion in porous media, and thin liquid film dynamics. One specific area which will be investigated is the regularity question in free-boundary problems arising from the degeneracy of certain non-linear parabolic equations, including the porous medium equation, the evolution p-laplacian equation and Gauss Curvature flow with flat sides. Another specific area of the main objectives concerns the study of ultra-fast diffusion parabolic equations, which have received large attention because of their relationship to differential geometry topics, such as the Ricci flow and the Yamabe flow. The solvability of the Cauchy problem for these equations, the study of the blow up profile of solutions, the nonradial structure solutions, and the uniqueness of solutions, are among the problems which will be investigated in this specific area. The still open question of the uniqueness of sign-solutions of the porous medium equation, in connection with the continuity question for a class of linear singular parabolic equations falls under the proposed major goals to be studied. The non-linear equations to be studied under this proposal form the basic concepts of many applications which deem to be important to technology and the society at large. The purification of materials, from chemicals to petroleum and even water, is often achieved by diffusion through filters. The purification filters are the porous media described in the proposal. Thin film dynamics and the Van der Waals forces operating between thin layers are described by singular quasilinear equations of ultra-fast d iffusion. The dynamics of population growth, polymer chain growth, including cross linking and high rate growth of biomolecules, are also non-linear phenomena amiable to our basic studies. The interesting problem of the expanding universe and other cosmological phenomena seem to be goverened by nonlinear dynamics, which in certain cases are applications of the more complex problems described here.