# Nonlinear Partial Differential Equations

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.

In summer session of 2014 students will pursue research on the open problem of evolution of interfaces
of solutions to the reaction-diffusion-convection equations.

Full solution of this problem for the reaction-diffusion equation is given in the following two papers by using significantly the theory of reaction-diffusion equations in general non-smooth domains developed in my JDE paper.

Introductory lectures on this topic will be posted soon.