Mathematical Sciences: Semilinear Elliptic Equations and Systems

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Work on this project combines geometrically motivated problems set in an analytic framework. It represents a continuation of research in four areas. In the first, work will continue in establishing existence and describing properties of conformal metrics with prescribed scalar or Gaussian curvature on complete manifolds. This is equivalent to finding positive solutions of an elliptic, nonlinear equation on the manifold, whose linear term is the Laplace-Beltrami operator in the metric with scalar curvature appearing as a multiplier of the unknown function. Although much is known about the compact case, relatively little has been done in the non-compact setting until recently. A second line of investigation will focus on developing a complete understanding of finite mass solutions and infinite mass solutions of the Matukuma equations in astrophysics. These are semilinear equations, first proposed in 1930, to model the dynamics of global star clusters. These equations are believed to be an improvement of Eddington's 1915 model. The unknown function here is the gravitational potential which, for the finite total mass case, has only recently been shown to have positive entire solutions (the corresponding Eddington equation has none). Many questions remain unanswered regarding the two models and discrepancies observed between their solutions. Work will also be done in determining whether the Lane- Emden equation of astrophysics has oscillatory as well as non- oscillatory solutions and analyzing the asymptotic behavior of singular radial solutions. In addition, relatively recent work on chemotaxis has led to new results on solutions of parabolic systems describing the oriented movement of cells in response to chemicals in the environment. Further work remains in determining how the cells move toward places of higher concentration and then aggregate.