RESEARCH

RESEARCH INTERESTS


Semipositone Problems
Superlinear problems on exterior domains.
  • $$ \left\{ \begin{array}{cl} - \Delta u = \lambda f(u), & x \in \Omega^c, \\ u=0, & x \in \partial \Omega, \\ u \to 0, & \|x\| \to \infty. \end{array} \right. $$
  • ✓ $f(0)<0$ (semipositone)
  • ✓ $\displaystyle \lim_{s \to \infty}\frac{f(s)}{s} = \infty$ (superlinear)
  • ✓ $\Omega$ is a bounded domain in $\mathbb{R}^n$.
  • ? existence, uniqueness, multiplicity of solutions
Nonlinear Boundary Conditions
Semipositone problems with nonlinear boundary conditions.
  • $$ \left\{ \begin{array}{cl} - \Delta u = \lambda f(u), & x \in \Omega^c, \\ \frac{\partial u}{\partial \eta} + c(u) u =0, & x \in \partial \Omega, \\ u \to 0, & \|x\| \to \infty. \end{array} \right. $$
  • ✓ $f$ is semipositone and superlinear.
  • ✓ $c:[0,\infty) \to (0,\infty)$ is continuous
  • ✓ $\Omega$ is a bounded domain in $\mathbb{R}^n$.
  • ? existence, uniqueness, multiplicity of solutions
Math Biology
Density dependent dispersal on the boundary
  • Modeling habitat surrounded by hostile matrix with nonlinear density dependent dispersal on the boundary using reaction-diffusion equations with nonlinear boundary conditions. (Single PDE)
  • Modeling competing species with nonlinear dispersal on the boundary based on density of competitor. (PDE systems)
  • ? existence, uniqueness, multiplicity, and stablity of steady states

COLLABORATORS



Dr. Jim Cronin

Louisiana State University

Dr. Jerome Goddard

Auburn University at Montgomery

Dr. Nsoki Mavinga

Swarthmore College

Dr. Catherine Payne

Winston-Salem State University

Dr. Dhanya Rajendran

Universidad de Concepcion

Dr. Stephen Robinson

Wake Forest University

Dr. Ratnasingham Shivaji

University of North Carolina at Greensboro

Dr. Inbo Sim

University of Ulsan

Dr. Byungjae Son

Wayne State University

LIST OF PUBLICATIONS

"Everyone engaged in research must have had the experience of working with feverish and prolonged intensity to write a paper which no one else will read or to solve a problem which no one else thinks important and which will bring no conceivable reward -- which may only confirm a general opinion that the researcher is wasting his time on irrelevancies."

Noam Chomsky