Mathematical Sciences: Spatial and Temporal Patterns in Non-Linear Differential Equations

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The investigators will continue their studies over the last several years of pattern formation in biological systems and the behavior of nonlinear equations in fluid mechanics. Their joint work includes extending a recent theorem which proves the existence of oscillatory solutions in an inhomogeneous reaction diffusion equation. They will also extend their recent studies of a fourth order nonlinear differential equation which arises in boundary layer theory in fluid mechanics. Professor Ermentrout will continue his joint research work with Professor Nancy Kopell of Boston University on the behavior of aggregates of coupled oscillators. He will also continue his joint research work on molluscan pigmentation patterns with Professor George Oster of University of California at Berkeley and work on fungal growth with Professor Leah Edelstein of Duke University. Professor Troy will extend his recent investigations of a reaction diffusion equation arising in chemical reactor and combustion theory. Depending on the values of physical parameters in the model, the solutions exhibit a variety of possible behavior ranging from finite time "blow-up" to finite time "extinction". He will continue his investigations of the behavior of self similar solutions which are used in analyzing the asymptotics of both the blow-up and extinction processes. This research falls into the general area of applied mathematical analysis, with potential applications to chemistry and biology.