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

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.