Analysis of the structure of solutions of nonlinear partial differential equations

研究了一维空间中由非线性抛物方程定义的无限维动力系统的结构，证明了它们的全局吸引子总是有限维流形。这一结果表明，解的长时间行为的本质特征可以用有限常微分方程组来描述，因此从定性理论的观点来看是重要的。值得注意的是，自20世纪80年代中期S作为研究吸引子有限维的工具而广为人知的所谓惯性流形理论，并不适用于我们所研究的方程。研究了退化扩散方程的解的性态，证明了任何不稳定的平衡解都存在无限Hausdorff维的不稳定流形。这一结果表明，退化扩散方程的动力学结构与非退化扩散方程的动力学结构有本质的区别。目前，Matano也在研究具有空间原系数的非线性扩散方程的行波性质，并取得了部分结果。研究了麦克斯韦方程和薛定谔方程相结合的数学模型。我们在解的唯一性和整体存在性方面得到了新的结果。4.本研究项目的研究者之一Kusuoka一直在用概率方法研究数学金融中的问题。他取得了一些有趣的结果。

Westudied the structure of the infinite dimensional dynamical systems defined by nonlinear parabolic equations in one space dimension and showed that their global attractors are always finite dimensional manifolds. This result implies that the essential features of the long-time behavior of solutions can be described by a finite system of ordinary differential equations, and is therefore important from the point of view of qualitative theory. It should be noted that the so-called inertial manifold theory, which has been well-known since mid 1980's as a tool for studying the finite dimensionality of attractors, does not apply to the equation treated in our research.2. We studied the behavior of solutions of degenerate diffusion equations and proved that any unstable equilibrium solution has an unstable manifold of infinite Hausdorff dimension. This result shows that there is essential difference between the dynamical structure of degenerate diffusion equations and that of nondegenerate equations. currently Matano is also studying the properties of traveling waves for nonlinear diffusion equations with spatially priodic coefficients and has obtained partial results.3. We studied a mathematical model which combines Maxwell equation and Schrodinger equation. We obtained new results on the uniqueness and global existence of solutions.4.Kusuoka, one of the investigaters of the research project, has been studying problems in mathematical finance with probabilistic method. He has obtained some interesting results.