Nonlinear Degenerate Parabolic Problems and Related Topics

摘要：本论文的主要目的是研究某些拟线性或全非线性退化抛物型方程与更复杂的微分几何问题，包括里奇流和高斯曲率流的关系。这些问题在诸如种群动力学、多孔介质扩散和薄液膜动力学等物理领域得到了应用。将研究的一个特定领域是由某些非线性抛物方程的退化引起的自由边界问题中的正则性问题，这些方程包括多孔介质方程、演化p-拉普拉斯方程和高斯曲率流。主要目标的另一个特定领域涉及超快速扩散抛物方程的研究，由于它们与微分几何主题（如Ricci流和Yamabe流）的关系，这些方程受到了广泛关注。这些方程的柯西问题的可解性、解的爆破剖面的研究、非径向结构解和解的唯一性等问题将在这一特定领域进行研究。多孔介质方程符号解的唯一性问题，以及一类线性奇异抛物方程的连续性问题，都属于本文提出的主要研究目标。在这个提议下要研究的非线性方程构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，甚至水的净化，通常是通过过滤器扩散来实现的。净化过滤器为提案中描述的多孔介质。薄膜动力学和作用于薄层之间的范德华力用奇异的超快流体准线性方程描述。人口增长的动态，聚合物链的增长，包括交联和生物分子的高速增长，也是非线性现象，有利于我们的基础研究。关于宇宙膨胀和其他宇宙现象的有趣问题似乎是由非线性动力学控制的，在某些情况下，这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.