
Research ProgramEvolution of Interfaces for the ReactionDiffusionConvection EquationsFaculty Mentor: Dr. Ugur AbdullaPartial 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 reactiondiffusionconvection 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.
In summer session of 2017 students will pursue research on the open problems of evolution of interfaces of solutions to reactiondiffusionconvection type equations. Introductory video lectures on this topic are available online here.References
