Can the solution to the heat equation be nonuniform? When considering heat in an otherwise constant body, it's a simple consequence of the second law of thermodynamics, as well as common sense that, given enough time, the temperature should be the same at all points in the body. However, according to a form of the heat equation $$\frac{\partial u}{\partial t} = \alpha \nabla ^2 u$$ where $\alpha$ represents the heat diffusivity, and $u(x,y,z,t)$ is a function that represents the temperature of a specific point, it should be possible to have an inconstant function, and still have zero heat flow! This is the case when the function is linear, such as $u(x,y,z,0) = x+y+z$ The laplacian of this function is zero, which means that the change in temperature with respect to time, or the heat flow, should also be zero. This then implies the temperature does not change, leaving the object with an indefinitely uneven heat distribution. It appears to me that the mathematical model and the physical world don't seem to agree in this situation. Am I missing something?
 A: You seem to be considering two different ideas: (1) the temperature at each point in the domain is constant, and (2) there is no heat flow through the domain. 
You can certainly make (1) true while (2) is false. Consider a slightly simpler example than the one you proposed: $$u(x,y,z,t) = x.$$ The heat flux at any point in the domain is given by $$Q(x,y,z,t) = -\nabla u,$$ or $$Q(x,y,z,t) = (-1, 0, 0),$$ i.e. a uniform non-zero heat flux in the $-x$ direction. 
For a finite domain, that means there also is a steady heat flow $Q$ through the boundary of the domain. 
There is nothing unphysical about that situation. For example consider the heat flow through a metal rod, with one end in boiling water and the other end in ice.
If you make the shape of the domain more complicated - for example, bend the rod in the previous example into an arc of a circle, or a helix, and make the cross section area of the rod non-uniform along its length - there can still be a steady state solution $U(x,y,z,t)$ which is independent of $t$. It will be the solution of $\nabla^2u = 0$ in the domain with the appropriate boundary conditions representing the physical system, including the heat flow into and out of the domain. 
Of course by the divergence theorem (which is equivalent to the conservation of heat energy in the domain!) the total heat flow into the domain for a steady state solution is always zero, but it is not zero at every point on the boundary.
