I'm currently working on trying to solve a system of PDE's of the form
$c_t=D_x(c_{xx} + c_{yy})+K_1 c + K_2 d$
$d_t= D_y(d_{xx}+d_{yy})+K_3 c + K_4 d$
that has "periodic boundary conditions" on a box $[0, 2\pi] \times [0, 2\pi]$, where the D_i and K_i are random constants (for now). I've searched extensively for the exact definition of this term, but only really came up with descriptions for a one variable ODE. I have relatively little experience with PDEs, but I understand that the boundary conditions will affect the solution to this system, so it's a phrase that I need to resolve.
Also, as a related question, I tried to solve this system based on a one-variable example using the guess
$c = c_0 e^{\sigma t} \cos(qx) \cos(ry)$
$d = d_0 e^{\sigma t} \cos(qx) \cos(ry)$
Is this the right idea? How will it interact with the "periodic boundary conditions" requirement?
Thanks for your help!
Periodic boundary conditions means that the solution is periodic as a function of each of the spatial variables. In other words $c(0,y)=c(2 \pi,y)$, $c(x,0)=c(x,2 \pi)$, similar for $d$, and a similar periodicity for the spatial derivatives. It has nothing to do with periodicity in time, which you will probably not have in this problem.
It can also be thought of as the solution living on a domain where the ends have already been glued together (which in 2D would yield the surface of a torus).
As for picking the right guess for the form of the solution, you're not really done yet. When the $K_i$ are zero, you have a heat equation in each variable separately (just with a possibly different diffusion coefficient). I suggest you look at example 5 of this: http://tutorial.math.lamar.edu/Classes/DE/SolvingHeatEquation.aspx