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The heat equation -- Fourier analysis and the energy method

We start with
The last a priori estimate shows that the parabolic solution becomes infinitely smoother than its initial data (- we "gain" infinitely many s-derivatives), and at the same time these higher derivatives decay faster as tex2html_wrap_inline11105.

Alternatively, we can work with the energy method. Multiply (para.1) by u and integrate to get
and in general
successive integration of (para.7) yields (para.5).

Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998