Research Description

Ralph E. Showalter joined the Mathematics Department at Oregon State University in the Fall of 2003. He left the University of Texas where he held the Blumberg Professorship in Mathematics. Since receiving the Ph.D. in Mathematics at the University of Illinois as an NSF Fellow, he has published about 90 research articles, one research monograph co-authored with R.W. Carroll, Singular and Degenerate Cauchy Problems , one graduate text, Hilbert Space Methods in Partial Differential Equations, one edited volume with J.T. Oden, Workshop on Existence Theory in Nonlinear Elasticity, and a recent volume in the Mathematical Surveys and Monographs of the American Mathematical Society, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. He contributed the chapter ``Micro-structure models of porous Media'' in the book Homogenization and Porous Media edited by Ulrich Hornung. His research interests include singular or degenerate nonlinear evolution equations and partial differential equations, related variational inequalities and free-boundary problems, and applications to initial-boundary-value problems of mechanics and diffusion. Among his technical contributions are the development of existence-uniqueness-regularity theory for pseudo-parabolic and Sobolev-type partial differential equations, existence theory of degenerate evolution equations, particularly the doubly-nonlinear cases. More applied contributions include the formulation and existence theory for Stefan free-boundary problems for a parabolic system, for the pseudo-parabolic equation, and for the hyperbolic telegraphers' equation. He introduced the fissured medium equation and the layered medium equation as models for diffusion in heterogeneous media and contributed to the development of distributed systems with microstructure and of hysteresis models. His current research interests are focused on the development of multi-scale models of coupled fluid-solid dynamics and flow in deformable porous media. Some of these are described in the survey article, Diffusion in Deforming Porous Media. A member of the American Mathematical Society, the Society for Industrial and Applied Mathematics, and the Texas Institute for Computational and Applied Mathematics, he has organized or co-organized a sectional SIAM meeting, an AMS special session, and an NSF Workshop on partial differential equations and applications. He regularly serves as referee for 20 research journals, and he is a member of the editorial boards of ten journals; he has supervised 18 Ph.D. dissertations.