Mathematical Sciences: Spatial and Temporal Patterns in Non-Linear Differential Equations

研究人员将在未来几年继续研究生物系统中的模式形成和流体力学中非线性方程的行为。他们的共同工作包括推广最近证明非齐次反应扩散方程振荡解存在的定理。他们还将扩展他们最近在流体力学边界层理论中出现的四阶非线性微分方程的研究。Ermentrout教授将继续与波士顿大学的Nancy Kopell教授共同研究耦合振子聚集体的行为。他还将继续与加州大学伯克利分校的George Oster教授共同研究软体动物的色素沉积模式，并与杜克大学的Leah Edelstein教授共同研究真菌的生长。特洛伊教授将扩展他最近在化学反应器和燃烧理论中出现的反应扩散方程的研究。根据模型中物理参数的值，解表现出从有限时间“爆炸”到有限时间“消失”的各种可能行为。他将继续研究自相似解的行为，这些解用于分析爆破过程和消光过程的渐近性。这项研究属于应用数学分析的一般领域，在化学和生物学方面有潜在的应用。

The investigators will continue their studies over the last several years of pattern formation in biological systems and the behavior of nonlinear equations in fluid mechanics. Their joint work includes extending a recent theorem which proves the existence of oscillatory solutions in an inhomogeneous reaction diffusion equation. They will also extend their recent studies of a fourth order nonlinear differential equation which arises in boundary layer theory in fluid mechanics. Professor Ermentrout will continue his joint research work with Professor Nancy Kopell of Boston University on the behavior of aggregates of coupled oscillators. He will also continue his joint research work on molluscan pigmentation patterns with Professor George Oster of University of California at Berkeley and work on fungal growth with Professor Leah Edelstein of Duke University. Professor Troy will extend his recent investigations of a reaction diffusion equation arising in chemical reactor and combustion theory. Depending on the values of physical parameters in the model, the solutions exhibit a variety of possible behavior ranging from finite time "blow-up" to finite time "extinction". He will continue his investigations of the behavior of self similar solutions which are used in analyzing the asymptotics of both the blow-up and extinction processes. This research falls into the general area of applied mathematical analysis, with potential applications to chemistry and biology.