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偏微分方程解的单调性和对称性。感兴趣的方程包括振动建模和非线性薛定谔方程中产生的方程。它们属于完全非线性椭圆型和抛物型方程。人们感兴趣的是理解规定最小周期的解或解不存在（爆炸）的奇点的性质。第一种情况已取得进展；要克服的主要障碍在于波算子的无限维核的复杂性。在过去的十年里，薛定谔方程解爆破的研究取得了很大的进展。已经证明，当空间域为星形时，解确实在有限时间内爆炸。这项工作将继续下去，以更好地理解奇点的本质，并获得关于爆炸可以预期的时间的更准确的估计。目前，许多相关的估计是非常粗略的。爆炸的开始还不知道有物理意义，因为最短的可能时间可能是指数级的。其他工作将研究由动力学和燃烧问题引起的行波解的存在性、唯一性和空间衰减性质。

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.