I asked to get the SS solution for non-linear parabolic PDE system:
$$\begin{align} \frac{\partial a}{\partial t}&=D \frac{\partial^2 a}{\partial r^2}+f(a,b)\\ \frac{\partial b}{\partial t}&=D \frac{\partial^2 b}{\partial r^2}+g(a,b) \end{align}$$
using $2$ different ways:
$1)$ PDE time solution and than check the solution when $t = \infty$
$2)$ ODE system that i get when set the time derivatives to zero. While the solution is stable (depend on the constants), i can find the SS easily using both methods.
When the solution in time does not converge to SS, the ODE still solvable and I got a solution (not the SS solution, because its not exist).
My questions are:
$1)$ How its come that there is solution for the ODE?
$2)$ What is the meaning of this solution?
** edited ** For example, my work dealt with the following problem:
$$\begin{align} \frac{\partial a}{\partial t}&=D \frac{\partial^2 a}{\partial r^2}+μ_0-ab^2-Ka\\ \frac{\partial b}{\partial t}&=D \frac{\partial^2 b}{\partial r^2}+ab^2+Ka-b\\ -1<r<1 \end{align}$$
The PDE Solution ,and the ODE Solution
There is no SS solution, but I still got solution from the ODE.
I used finite differences methods for solving the problems.
Thanks