Mathematical Methods in Biology and Neurobiology by Jürgen Jost

Author:Jürgen Jost
Language: eng
Format: epub
Publisher: Springer London, London

(4.1.86)

Here, the are the orthonormal eigenfunctions of ,

and with

We then have an expansion for the solution of

(4.1.87)

namely,

(4.1.88)

Since all the are nonnegative, we see from this representation that all the “modes” of the initial values are decaying in time for a solution of the heat equation. In this sense, the heat equation regularizes or smoothes out its initial values. In particular, since thus all factors are less than or equal to for , the series (4.1.88) converges in , because (4.1.86) does.

If we write

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