Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 2

Apply separation of variables with $\rho (x,t)=X(x)T(t)$ and temporal factor $T=e^{-\lambda t}$. The spatial eigenvalue equation isThe nontrivial solutions of these Dirichlet boundary conditions are sine eigenfunctions. Normalizing in the usual inner product on $(0,L)$ giveswith ${\int }_0^L{\psi }_n{\psi }_{m}dx={\delta }_{nm}$. The superposition principle and the Fourier sine series then giveThe constants are the initial-data coefficients $a_n={\int }_0^L\rho (x,0){\psi }_n(x)dx$. Here physical diffusion means $D>0$.

By orthogonality and the sampling property of the Dirac delta distribution,Consequently the reaction-diffusion equation solution isFor every $t>0$ this eigenfunction expansion is smoothed by the Gaussian decay in $n$. As $t\downarrow 0$, it recovers the specified initial condition in the sense of distributions.

The smallest decay-rate eigenvalue is ${\lambda }_1=D{\pi }^2/L^2-\gamma$, so the threshold isBecause $0<x_0<L$, $a_1>0$. For every fixed interior point $0<x<L$, the first sine eigenfunction is positive and the remaining terms have strictly faster decay relative to it. HenceOne can justify the relative-error limit directly: after factoring out the first exponential, the absolute remainder is bounded by $(2{\rho }_0/L){\sum }_{n\ge 2}{e}^{-D{\pi }^2(n^2-1)t/L^2}$, which tends to zero. Thus $\boxed{\rho \to 0\text{ if }\gamma <{\gamma }_c}$, and $\boxed{\rho (x,t)\to +\infty \text{ for each interior }x\text{ if }\gamma >{\gamma }_c}$. The boundary values stay zero. At equality the first term is stationary and the density approaches its positive sine profile.

Let $f(x)=\rho (x,0)$. The first normalized Fourier sine series coefficient isStrict positivity follows because $f\ge 0$, has positive total mass, and $\sin (\pi x/L)>0$ everywhere inside the interval. In particular, there is no cancellation that could remove the slowest term. This is the positivity of the first Dirichlet reaction-diffusion coefficient.

For finite initial mass $M={\int }_0^{L}f(x)dx$, all coefficients satisfy $|a_n|\le \sqrt{2/L}M$. The eigenfunction expansion therefore gives the uniform boundThe positive first profile thus controls the long-time behavior. We recover exactly the threshold $\boxed{{\gamma }_c=D{\pi }^2/L^2}$, with decay below it and unbounded positive growth at every interior point above it. The finite-mass condition is the usual meaning of an initial density profile here; standard positive-time smoothing gives the same conclusion in the usual broader admissible data classes.