U.S.-Austria Cooperative Research: Volterra Integral Equations and Applications

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This award will support a two-year collaborative research effort in mathematics involving Professor Ronald C. Grimmer, Southern Illinois University at Carbondale; Professor Richard K. Miller, Iowa State University; Professor Wilhelm Schappacher, University of Graz, Austria; and Dr. Wolfgang Desch, also at the University of Graz. The cooperating mathematicians are studying a number of related problems involving so-called Volterra integrodifferential equations in Banach space. Their research will deal with various questions concerning control and qualitative behavior of solutions of partial differential integral equations. Of particular interest will be determining the behavior of solutions which have singularities in the initial or boundary data. It is important to determine if these singularities propagate and, if so, additionally to determine the behavior of these singularities. Included also will be the study of asymptotic behavior of solutions especially as it is related to control problems. The expected results of this research will be largely of an abstract mathematical nature, but the work may well lead to mathematical techniques which are applicable to practical problems, as for example in the study of the physical behavior of materials having both viscous and elastic properties. Examples of such viscoelastic media are polymeric solids, high molecular weight polymer fluids, blood, and fluidized coal. Another application of the equations being studied here is to electrodynamics. The mathematicians collaborating in this project have both similar and complementary strengths and interests. They have worked together productively in the past, and this award will permit them to continue their mutually beneficial cooperation.