Mathematical Sciences: Structural Stability for Parabolic Equations

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9400233 Lu This award supports mathematical research on the development of structural stability for parabolic partial differential equations and related problems. The structural stability of dynamical systems is one of the fundamental problems in the theory of dynamical systems. One of the main problems currently facing researchers in the area concerns infinite dimensional dynamical systems on locally non-compact spaces. While much progress has been made, those systems generated by parabolic equations define only semiflows since backwards solutions may not exist and the phase spaces are not locally compact. This work will first focus on structural stability for scalar equations with periodic boundary conditions. In particular efforts will be made to understand stability in a neighborhood of an attractor and to study the quasiperiodic and almost periodic scalar parabolic equations. Involved in this work is the skew product flow, the lap number and Floquet theory. Other studies of Floquet theory for time dependent parabolic equations and smooth conjugacy for parabolic equations will also be carried out. Partial differential equations form a basis for mathematicalmodeling of the physical world. The role of mathematicalanalysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, for mathematical models to adequately describe reality, the qualitative properties of solutions should not change significantly when the model is subjected to small, allowable perturbations. An attempt to formalize this point of view leads to the concept of structural stability. ***