Mathematical Sciences: Elliptic and Parabolic Systems of Nonlinear Equations

该奖项为两个当前热门问题的数学研究提供了支持，即扩散系统中行波的稳定性问题和无界圆柱域中的对称破缺分岔问题。在第一个实例中，我们将分析与生态学、燃烧理论和神经生理学中特定模型方程相关的偏微分方程的反应-扩散系统的行波解的稳定性。该技术将包括努力理解线性化抛物系统的特征值分布的行波解。然后可以应用一般拓扑方法来深入了解真解的定性性质。目前，人们对这类方程组所知甚少。最初的工作将包括研究重要的具体例子和采用新的工具。第二项研究与第一项有关，因为我们将研究相应的椭圆型稳态方程的解。这里的问题集中在解决方案分支的分叉上。具体地说，工作将在无限圆柱体（取决于参数）中定义的半线性方程上完成，这些方程承认正径向解。非径向解随着参数的变化而分叉。有几个问题与对称性破缺有关。一是理解通过将问题简化为常微分方程组而得到的相关向量场的继承对称性。第二种是考虑将圆柱体的几何形状从圆形变为椭圆形的影响，最后，人们希望找到比预期的周期分支更复杂的分支。这项研究在应用科学的许多领域有潜在的用途，在这些领域中，反应扩散和纳维-斯托克斯方程被用于建模。它也可能导致对标量扩散方程解的混沌动力学的新见解。

This award provides support for mathematical research on two issues of current interest, the question of stability of travelling waves in diffusive systems and symmetry breaking bifurcations in unbounded cylindrical domains. In the first instance, work will be done analyzing the stability of travelling wave solutions of reaction-diffusion systems of partial differential equations related to specific model equations in ecology, combustion theory and neurophysiology. The technique will involve efforts to understand the distribution of eigenvalues of the linearized parabolic systems about a travelling wave solution. General topological methods may then be applied to gain insight into the qualitative nature of the true solution. At the present time very little is known about systems of such equations. Initial work will involve studies of important specific examples and bringing new tools to bear. The second line of investigation is related to the first in that solutions of the corresponding steady-state equations, which are elliptic, will be studied. Questions here focus on the bifurcation of branches of solutions. Specifically, work will be done on semilinear equations defined in infinite cylinders (depending on a parameter) which admit positive radial solutions. Non-radial solutions bifurcate out as the parameter varies. Several questions arise in connection with the symmetry breaking. One is to understand the inherited symmetries of the associated vector field obtained by reduction of the problem to systems of ordinary differential equations. A second is to consider the effects of changing the geometry of the cylinder from circular to elliptic and finally, one wants to find more complex bifurcations than the expected periodic ones. This research has potential for use in many areas of applied science where reaction-diffusion and Navier-Stokes equations are used in modeling. It may also lead to new insights into chaotic dynamics in solutions of scalar diffusion equations.