conservation of total heat

Pour a fixed amount of dye into a sealed tank of water. The dye spreads and fades to a faint even tint, but the total amount of dye in the tank never changes — none was created, none destroyed, it just got rearranged. Conservation of total heat is the same accounting fact for the heat equation: the total amount of heat (or substance) is preserved as it diffuses, as long as nothing crosses the boundary.

Take the heat equation u_t = k u_xx and integrate both sides over the whole interval or line. The total heat is Q(t) = integral of u(x,t) dx. Differentiating under the integral, dQ/dt = integral of u_t dx = integral of k u_xx dx = k [u_x] evaluated at the endpoints. So the rate of change of total heat equals the net flux through the boundary, by the fundamental theorem of calculus. If the boundary lets no heat escape — insulated (Neumann) ends u_x = 0, or the whole line where u and u_x vanish at infinity — then dQ/dt = 0 and Q is constant for all time. The total is conserved. This is exactly why the heat kernel has area 1 for every t: it represents one unit of heat, redistributed but never lost.

Conservation here is not a separate postulate; it is built into the equation, which was derived from a conservation law in the first place (rate of accumulation = minus divergence of flux). It gives you a free check on any solution and a quick qualitative grip: with insulated walls the bar drifts to the uniform temperature equal to its average initial temperature, because that is the only constant profile carrying the same total heat. Caveat: conservation requires no flux through the boundary and no sources. With Dirichlet ends held cold, heat leaks out and the total decays to zero; with a source term (an inhomogeneous equation) heat is added, and you track the balance with Duhamel's principle.