Heat equation - stationary solution

What are the conditions for the solution of the heat equation to become [sufficiently] stationary, i.e. to not change any more over time?

Here, I am talking about the 3D heat equation with Neumann and Dirichlet boundary data: $$\partial_t u - a \Delta u = f$$ with initial condition $$u(x,0) = u_0(x)$$ and boundary conditions $$u = g$$ and $$\partial_n u = q$$ on some part of the boundary each.

Let's assume that $$f,u_0,g,q$$ and the unspecified domain are "sufficiently nice" for our problem. I am more looking for a broad answer regarding the convergence to the steady-state solution, as I observed this behaviour during some experiments.

Some thoughts of mine so far:

I guess that $$f,g,q$$ cannot change over time, to allow a steady-state solution? Or do I even need some form of $$q=0$$?

Or is this a simple matter of solving the new elliptic problem resulting from $$\partial_t u = 0$$

A followup question would be: Can one do a time estimate for when such a solution might be reached?

• You are correct that $f,g, q$ must all become constant with respect to time, since all three can be expressed in terms of $u$, if $u$ is not time-dependent, they cannot be either. I suspect that is also a sufficient condition. If there is nothing exciting the system, I would expect it to enter a steady-state. But I don't have a mathematical demonstration. – Paul Sinclair Mar 16 at 20:21