This is what happens in an explosion. The question of when heat will be able to diffuse away and when it will not has obvious practical importance. The boundary condition for this elliptic equation is that, at the container's surface, T should equal the temperature of the container. We have developed a simple technique for answering this question of existence or nonexistence for such equations. For example, when fuel is being injected into a cylindrical pipe and is flowing along the pipe, my collaborators G.I. Barenblatt and A.J. Chorin and I have shown that explosion may or may not occur depending on the relationship between the radius and the length of the pipe.
Explosion is more likely in longer, wider pipes than in shorter, narrower ones.
If a pipe is narrow enough, explosion will not occur no matter how long it is. Here is a graph of the critical length of a pipe versus its radius, assuming that the reacting mixture flows smoothly. When the radius is increased, the critical length becomes shorter according to a power law in the radius. We discovered this power law, and a related law that applies when the pipe flow is turbulent. Professors Barenblatt and Chorin have also studied the generalization of these ideas to turbulent flow.