## Solving for homogenous steady states

Wasn't sure where to post these questions..

1. In simple diffusion, concentration c(x,t) obeys:
dc/dt = D(d^2c/dx^2)

Show mathematically that if a chemical is diffusing inside a closed container that it eventually reaches the same concentration everywhere by solving for the steady state and showing that it is spatially uniform.

So I know that solving for the steady state involves setting the time derivative to 0, and if it's spatially uniform, the space (dc/dx) derivative also to 0. But would I show this?

2. For the following sets of reaction terms, R1(c1,c2) and R2(c1,c2), determine whether or not a homogeneous steady state can be obtained and whether the system is capable of giving rise to diffusive instability. If so, give explicit conditions for instability to arise, and determine which perturbation frequency ("mode") would be most destabilizing:

a) Lotka-Volterra:
R1 = a(c1) - b(c1)(c2)
R2 = -e(c2) + d(c1)(c2)

b) Glycolic Oscillator:
R1= delta - k(c1) - (c1)(c2^2)
R2 = k(c1) + (c1)(c2^2) - (c2)