Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 29C / iii

Table of contents
- Solution iii

Solution

The Fourier coefficients obey

\overset{u}{˙}_{n} + (1 + n^{2}) u_{n} = f_{n}

, so

u (t, x) = n \in Z \sum [\frac{f _{n}}{1 + n ^{2}} + ((u_{0})_{n} - \frac{f _{n}}{1 + n ^{2}}) e^{- (1 + n^{2}) t}] e^{in x} .

The coefficient decay of

u_{0}, f

and the additional exponential damping justify all derivatives for

t > 0

and recovery of the initial data. Thus this is a smooth periodic solution. Every transient mode decays, giving

u (t, \cdot) \to u_{f}

uniformly, indeed with all spatial derivatives. The elliptic solution in part (i) is the attracting steady state of this forced heat equation.

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