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?

  • $\begingroup$ 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. $\endgroup$ – Paul Sinclair Mar 16 at 20:21

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