the steady-state heat distribution

Leave any heated object alone long enough, with its surroundings held fixed, and it stops changing — it settles into a final temperature pattern that does not move. That final, unchanging profile is the steady-state heat distribution. It is what the heat equation is heading toward as time goes to infinity.

Steady state means u no longer depends on time, so u_t = 0. Put that into the heat equation u_t = k u_xx and you get k u_xx = 0, that is u_xx = 0 in one dimension, or Laplacian u = 0 in higher dimensions — Laplace's equation. So the steady state of the heat equation is a harmonic function, satisfying the same boundary conditions as the original problem but with the time dependence switched off. In one dimension u_xx = 0 means u is a straight line, so a bar with ends held at temperatures a and b settles into the linear profile interpolating between them. In two or three dimensions you must solve the Dirichlet (or Neumann) problem for Laplace's equation with the prescribed boundary data. The steady state is the harmonic limit of diffusion.

This connects two of the model PDEs: the elliptic Laplace equation is exactly the long-time, time-independent limit of the parabolic heat equation. The decay of Fourier modes makes this concrete — every time-dependent mode dies like e^(-k n^2 t), so only the steady part survives as t grows. Conditions for a steady state to exist and be unique come from the boundary type: Dirichlet data gives a unique harmonic steady state; pure Neumann (fully insulated) needs the heat input to balance (a compatibility condition) and then the steady state is determined only up to an additive constant, fixed by conservation of the initial total heat. Caveat: a steady state is reached only if the boundary data and sources are themselves time-independent; with time-varying forcing the solution may never settle.