Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 6B / a / Solution

Integrating the population equation over the line and using vanishing boundary flux gives $\dot{M}=-\mu M$, where $M(t)=\int n(x,t)dx$. Since $M(0)=Q$, one has $M(t)=Q{e}^{-\mu t}$. On the other hand the similarity substitution $x=\lambda (t)\eta$ givesConsequentlyThe decay condition is on the nonlinear diffusive flux $n{n}_x$; compact support also satisfies it.