Mathematical Sciences: On the Existence and Qualitative Properties of Solutions of Nonlinear Partical Differential Equations

这项工作的重点将是研究单调性和 偏微分方程解的对称性 方程 感兴趣的方程包括以下方程： 振动建模和非线性薛定谔方程。 他们 落入被称为完全非线性椭圆的类别， 抛物方程 一个是有兴趣了解 规定了最短期限的解决办法或 解不再存在的奇点（爆破）。 第一个案件已经取得进展; 我们要克服的是无限维的复杂性 波算子的核 爆破研究进展 薛定谔方程的解已经提出了 在过去的十年里，这一数字大幅上升。 已经表明，当 空间域是星形的，解确实在有限的 时间 将继续这方面的工作，以更好地了解 性质的奇异性，并获得更尖锐的估计 关于爆炸的时间 目前，许多相关的估计是非常粗略的。 的发作 爆炸还不知道有物理意义，因为 最小可能时间可以是指数级的大。 其他工作将调查的存在性，独特性和 行波解的空间衰减性质 动力学和燃烧问题。

The focus of this work will be the study of monotonicity and symmetry properties of solutions of partial differential equations. Equations of interest include those arising in modeling vibrations and in nonlinear Schrodinger equations. They fall into the classes known as fully nonlinear elliptic and parabolic equations. One is interested in understanding solutions where the minimal period is prescribed or the nature of the singularity where a solution ceases to exist (blow up). Progress in the first case has been made; the major obstacle to be overcome rests with the complexity of the infinite dimensional kernel of the wave operator. Development in the study of blow up of solutions of the Schrodinger equation have advanced considerably in the past decade. It has been shown that when the spatial domain is star-shaped, solutions do blow up in finite time. Work will continue in this vein to understand better the nature of the singularity and to obtain sharper estimates regarding the time at which the blow up can be expected. Currently, many related estimates are very crude. The onset of blow up is not yet known to have physical significance because the minimal possible time may be exponentially large. Other work will investigate the existence, uniqueness and spatial decay properties of solutions of traveling waves arising from dynamics and combustion problems.