Transient diffusion with compact support throughout—not just initially

[An update made on 2 January 2013 appears near the end]

The following is the question I raised today at the Mathematics Stack Exchange [^]. (It is only today that I became a member there.)

The Question:

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}.

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I was expecting answers to affirm instantaneous action at a distance. The first full answer to arrive [^] confirms this anticipation. However, it’s too late in the night to go and point out the Brownian movement-related objection to it. I will do it some time tomorrow. (It’s a mathematics forum. I am not too comfortable writing maths-related comments directly. I have to first translate my thoughts from physics to mathematics.) Also, tomorrow, I will come back here and decide on the fly whether to add the usual final section “A Song I Like,” or not. Also, I will add the tags to this post, tomorrow.

In the meanwhile, think about it and see if you wish to answer or have an interaction on this topic, at Math[s] StackExchange.

* * * * *   * * * * *   * * * * *

Update on 2 January 2013: I have now begun interaction at the Maths StackExchange forum. Also, now am adding the following section (even if I still go jobless), and the post tags. Guess I will have to be back very soon with another post on this same topic. A sort of “layman’s” (at least an engineer’s) version of the problem, and my position about it. I often find that by loosening the demands of the mathematical rigor a bit (though not the rigor of the basic logic or thought), a layman’s version helps bring in more context more easily, and thereby is actually helpful in delineating issues—both problems and solutions—better. So I will attempt doing that in a next post.

A Song I Like:
(Hindi) “swapn jhade phool se, mit chubhe shool se…” (“karvan guzar gaya, gubaar dekhate rahe”)
Lyrics: Neeraj
Singer: Mohammed Rafi
Music: Roshan

[E&OE]

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2 thoughts on “Transient diffusion with compact support throughout—not just initially

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