  The dynamics of classical $\phi^4$ theory under weak and strong thermal
gradients is studied. We obtain the thermal conductivity of the theory
including its temperature dependence. Under moderately strong thermal
gradients, the temperature profiles become visibly non-linear, yet the
phenomenon can be understood using the linear response theory. When we move
further away from equilibrium, we find that the linear response theory
eventually breaks down, and the concept of local equilibrium also fails.
