## Research Program

### Evolution of Interfaces for the Reaction-Diffusion-Convection Equations

Faculty Mentor: Dr. Ugur Abdulla

Partial differential equations (PDEs) are central to mathematics, whether pure or applied. They arise in mathematical models of real world problems, where dependent variables vary continuously as functions of several independent variables, usually space and time. PDEs arising in a majority of real world applications are nonlinear. Despite its complexity, in the theory of nonlinear PDEs one can observe key equations or systems which are essential for the development of the theory for a particular class of equations. One of those key equations is the nonlinear reaction-diffusion-convection equation:

$u_t=(u^m)_{xx}+bu^{\beta}+c(u^{\gamma})_x,$

where $u=u(x,t),x\in \mathbb{R}, t>0$, $b,c \in \mathbb{R}$, $m>0, \beta>0, \gamma>0$. It arises, for instance, in the theory of gas flow through a porous medium, heat radiation in plasmas, spatial spread of biological populations, diffusion of particles in plasma etc.
This project will provide undergraduate students with an effective introduction to some crucial concepts and methods of nonlinear science, such as the existence of free boundaries and the occurrence of regularity thresholds. Today, there exists a complete theory covering the well-posedness of boundary value problems in general domains, comparison theorems, and the interior and boundary regularity of solutions. As for the qualitative properties of solutions to degenerate and singular parabolic equations, there are many important open problems.

In summer session of 2017 students will pursue research on the open problems of evolution of interfaces of solutions to reaction-diffusion-convection type equations.