Editing Heat equation (section)

=== Character of the solutions ===
[[Image:Heatequation exampleB.gif|right|frame|Solution of a 1D heat partial differential equation. The temperature (<math>u</math>) is initially distributed over a one-dimensional, one-unit-long interval (''x''&nbsp;=&nbsp;[0,1]) with insulated endpoints. The distribution approaches equilibrium over time.]]

[[File:Heat Transfer.gif|thumb|The behavior of temperature when the sides of a 1D rod are at fixed temperatures (in this case, 0.8 and 0 with initial Gaussian distribution). The temperature approaches a linear function because that is the stable solution of the equation: wherever temperature has a nonzero second spatial derivative, the time derivative is nonzero as well.]]

The heat equation implies that peaks ([[local maximum|local maxima]]) of <math>u</math> will be gradually eroded down, while depressions ([[local minimum|local minima]]) will be filled in.  The value at some point will remain stable only as long as it is equal to the average value in its immediate surroundings.  In particular, if the values in a neighborhood are very close to a linear function <math>A x + B y + C z + D</math>, then the value at the center of that neighborhood will not be changing at that time (that is, the derivative <math>\dot u</math> will be zero).

A more subtle consequence is the [[maximum principle]], that says that the maximum value of <math>u</math> in any region <math>R</math> of the medium will not exceed the maximum value that previously occurred in <math>R</math>, unless it is on the boundary of <math>R</math>. That is, the maximum temperature in a region <math>R</math> can increase only if heat comes in from outside <math>R</math>. This is a property of [[parabolic partial differential equation]]s and is not difficult to prove mathematically (see below).

Another interesting property is that even if <math>u</math> initially has a sharp jump (discontinuity) of value across some surface inside the medium, the jump is immediately smoothed out by a momentary, infinitesimally short but infinitely large rate of flow of heat through that surface. For example, if two isolated bodies, initially at uniform but different temperatures <math>u_0</math> and  <math>u_1</math>, are made to touch each other, the temperature at the point of contact will immediately assume some intermediate value, and a zone will develop around that point where <math>u</math> will gradually vary between <math>u_0</math> and <math>u_1</math>.

If a certain amount of heat is suddenly applied to a point in the medium, it will spread out in all directions in the form of a [[diffusion wave]].  Unlike the [[mechanical wave|elastic]] and [[electromagnetic wave]]s, the speed of a diffusion wave drops with time: as it spreads over a larger region, the temperature gradient decreases, and therefore the heat flow decreases too.