Research Bio

Ralph Showalter joined the Mathematics Department at Oregon State University in the Fall of 2003. Previously he held the Blumberg Centennial Professorship in Mathematics at the University of Texas at Austin and was a founding member of the Oden Institute for Computational Engineering and Sciences (previously ICES). He was also a visiting professor at Brown University, Virginia Tech, Purdue University, and University of Augsburg. Ralph served as OSU Mathematics Department Chair in 2004-2007. He also co-founded the Applied and Computational Mathematics Seminar series at Oregon State in 2004 which he continues to co-organize. A member of the Society for Industrial and Applied Mathematics, he co-initiated the Texas Differential Equations Conference series in 1978, and co-organized a sectional SIAM meeting, an AMS Western section meeting in 2017, an NSF Workshop on Applications of Functional Analysis in Mechanics in 1977, a DOE-NSF Workshop on "Modeling, Analysis and Simulation of Multiscale Nonlinear Systems" in 2007, the inaugural Cascade RAIN meeting in 2014, and SIAM Pacific Northwest Section in 2017, as well as numerous sessions at SIAM meetings. He is on the editorial boards of ten journals, co-editor of four special volumes, and has served as referee for 20 research journals.

Since receiving the Ph.D. in Mathematics at the University of Illinois as an NSF Graduate Fellow, Ralph authored or co-authored over a hundred research articles and three research monographs and supervised 22 Ph.D. dissertations. His research was supported by the National Science Foundation 1972-2000, the Office of Naval Research 1988-1991, and by the Department of Energy in 2005-2010, including the DOE Multiscale Pacific Northwest Consortium. 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. His current work is on the analysis of multi-scale models of coupled fluid-solid dynamics, flow in deformable porous media, and hysteresis.

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 and nonlinear systems in mixed form. 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, the phase-change problem of advection of methane in the hydrate zone, quasi-static Biot systems of poroelasticity, and the coupled Biot-Stokes system. 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 of partial differential equations with microstructure, and those with hysteresis.