Transient diffusion with compact support throughout (not just initially) [Pardon my lack of rigor; I am an engineer by training. Also, for convenience, allow me to make this question as concrete as possible.]
Assume the simplest linear diffusion equation: $\alpha \dfrac{\partial^2u}{\partial x^2} = \dfrac{\partial u}{\partial t}$, where $u$ is the temperature and $\alpha$ is the thermal diffusivity. 
The domain is finite, say, $[-100, 100]$. (If the assumption of an infinite domain makes it possible (or more convenient) to answer this question, then please assume so. However, the question of interest primarily pertains to a finite domain.) 
Assume that the initial temperature profile has a compact support, say over $[-1, 1]$. 
After the passage of an arbitrarily small but finite duration of time:
(i) would the temperature profile necessarily have support everywhere over the entire domain?
(ii) or, is it possible that a solution may still have some compact support over some finite interval that is smaller than the whole domain?
Can it be proved either way? Given the sum totality of today's mathematics (i.e. all its known principles put together), is it possible to pick between the above two alternatives in general?
A subsidiary question only if the alternative (ii) is possible: please supply an example, better so, it is of a kind wherein the initial profile is infinitely differentiable, e.g. the bump function $e^\frac1{x^2-1}$.
Thanks in advance.
--Ajit
[E&OE]
 A: The classical heat equation exhibits infinite propagation speed, unlike the wave equation. For every domain $\Omega$, such as $[-100,100]$, there is an associated heat kernel $\Phi_\Omega(x,y,t)$ from which the solution (with zero boundary data) is obtained as 
$$u(x,t)=\int_\Omega u(y,0)\,\Phi(x,y,t)\,dy$$
An explicit form of $\Phi$ is available for certain domains, e.g., Giuseppe Negro gave it for the line. But no matter what the domain, for any fixed $t>0$ all values of the kernel $\Phi$ are strictly positive. This implies that if $u(x,0)$ is positive on some part of the domain and zero elsewhere, the solution $u(x,t)$ will be positive on all of the domain. 
Infinite propagation is not physically realistic, but is a consequence of mathematical process of abstraction in which a finite number of molecules of positive mass are replaced by infinite number of massless points on a line. In other words, it is built into the heat equation. If one wishes to model finite propagation speed, there are two options: 


*

*change the equation. The recent article Some diffusion equations with finite propagation speed by F. Andreu, V. Caselles, J.M.Mazon, and S. Moll gives an overview of such efforts. The key term here is flux-limited diffusion. Unfortunately, there is a steep price to pay: such modified equations are a lot harder to deal with (they are nonlinear).

*keep the equation but replace the notion of "support". Namely, we may decide that the increase of temperature, say, by $10^{-3}$ of a Celsius degree is not physically important. (This threshold should be picked based on the details of the problem.) Then define the "substantial" support of the solution at time $t$ as the region where $u(x,t)>10^{-3}$. This set will not immediately expand to the entire domain.

