Analysis of the structure of solutions of nonlinear partial differential equations

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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.