Mathematical Sciences: Elliptic and Parabolic Systems of Nonlinear Equations

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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.