I just started studying different types of PDEs and solving them with various boundary and initial conditions. Generally, when working on class assignments the professors will somewhat lead us to the answer by breaking a single question (solving a PDE) into parts and starting with things like: $(a)$ start by finding the steady-state solution, $(b)$ ….
My question, however, is rather general. I want to know under what conditions does a steady-state solution exist for a PDE. If possible, I want to keep the question general, but I know it helps to have a reference sometimes so consider the following PDE:
$$ \frac{\partial c_1}{\partial t} = \frac{\partial^2}{\partial z^2}\left[\frac{\partial^2 c_1}{\partial z^2}-\frac{\partial g(c_1)}{\partial c}\right]+F $$
where it's known that $ c_1 = c_1(z,t)$ only. Is it reasonable to try to solve for a steady-state solution and let $c_1(z,t) = s(z) +v(z,t)$? Would it depend on the associated boundary conditions? Is the existence of such a solution only dependent on the properties of the PDE? like it being linear? Any help, links, or comments would be much appreciated.
Again, if the example is ignored that's totally fine, the answer desired is the most general one possible!
Best Answer
A steady state for $$ u_t = f(u,u_x,\ldots) $$ exists if there is a solution to $$ 0 = f(u,u_x,\ldots), $$ which is an ordinary differential equation when there is only onoe space variable $x$. That solution needs to satisfy any given boundary conditions to be correct. But to be useful we hope it has the property that other solutions are attracted to it as time increases. The function $f$ need not be linear.
In your example, maybe you meant to write $\frac{\partial g(c_1)}{\partial c_1}$in which case that is a possibly nonlinear function of $c_1$. Any constant $c_1$ that makes that zero is a steady state solution if also $F = 0$. Whether it attracts other solutions would take a bit of study.
It is not usually going to work to look for $c_1 = s(z)+v(z,t)$ unless the equation is linear. Boundary conditions do matter, but the main point of steady states, when attractive, is that initial conditions don't matter in the long run.
This question leads to other ideas too. For example a forced damped oscillator $$ y''(t) +hy'(t)+ky(t) = f(t) $$ doesn't have a steady state, but has the property that the difference of any two solutions tends to zero as $t$ goes to infinity. That is rather significant, and also says that initial conditions don't matter in the long run.
Another related idea is the traveling wave solutions, $$ c_1(z,t) = f(z+kt)$$ that some PDEs have, where $k$ is a constant wave speed to be determined, and $f$ satisfies on ordinary differential equation.