Mathematical Sciences: Nonlinear Functional Analysis in Applied Science

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9401418 Takac This project is concerned with development and applications of methods of modern functional analysis to problems arising in a number of generally well-accepted mathematical models in applied science. The investigator proposes to continue his development of modern functional analytic techniques that are suitable for studying large-time asymptotic behavior of solutions to monotone dynamical systems. A particular emphasis of this part of the proposed research will be on the large-time asymptotic behavior of orbits (trajectories) in almost periodic monotone dynamical processes. Various instability phenomena are described by a parabolic system of complex Ginzburg-Landau equations. The investigator proposes to establish the local (in time) existence, uniqueness and complex analytic extension of solutions to the parabolic system. This will be followed by a development of suitable global estimates in the Lebesgue-Hardy spaces over a region in multi-dimensional complex space in order to prevent all the derivatives of the solution from blowing-up in finite time. For the Ginzburg-Landau equation it should be possible to obtain such estimates directly from the equation. Global existence, uniqueness and complex analytic extension of solutions to a mathematical model guarantee the existence and uniqueness for all times of a particular physical (biological, material or economic) structure described by this model, and its smooth dependence upon the spatial and temporal variables (no sudden jumps). These properties are essential for the design of measurements to be performed on the physical structure. Large- time asymptotic behavior of solutions to a model determines the ultimate state of relationships within the modeled structure. Mathematical models investigated in this project are expressed in terms of non-linear differential equations. ***