Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 29C (Partial Differential Equations)

Write the Fourier series $f(x)={\sum }_{n\in \mathbb{Z}}{f}_n{e}^{inx}$, with $f_n=(2\pi {)}^{-1}{\int }_{-\pi }^{\pi }f(x)e^{-inx}dx$. Smoothness makes its coefficients decay faster than every inverse power. Settingtherefore gives a smooth periodic function, and termwise differentiation proves $-u_f^{\prime \prime }+u_f=f$. Conversely every solution must have these coefficients, proving uniqueness. Equivalently, a homogeneous solution has $\int (|u^{\prime }{|}^2+|u{|}^2)=0$ by integration by parts, so vanishes.

Expand the quadratic functional and use periodic integration by parts:The first integral is zero because $-u_f^{\prime \prime }+u_f=f$. Thus the difference is $\frac{1}{2}\int ({\phi }^{\prime 2}+{\phi }^2)>0$ for every nonzero $\phi$. $u_f$ is the unique strict minimizer.

The Fourier coefficients obey ${\dot{u}}_n+(1+n^2)u_n=f_n$, soThe 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 $\boxed{u(t,\cdot )\to u_f}$ uniformly, indeed with all spatial derivatives. The elliptic solution in part (i) is the attracting steady state of this forced heat equation.

Differentiate under the integral and use the evolution equation:Thus $\boxed{I_f[u(t)]\le I_f[u(s)](t>s>0)}$. This is a gradient flow of the periodic screened Poisson energy. Part (iii) gives convergence in the function and its first derivative, so $\boxed{I_f[u(t)]\to I_f[u_f]}$, the minimum from part (ii).