Morphogen reaction-diffusion equation

A morphogen reaction-diffusion equation models a spatial concentration by diffusion and local production or decay. Linearization at a homogeneous equilibrium $C=0$ replaces $f(C)$ by $f^{\prime }(0)C$.

The eigenfunctions satisfying $X(0)=0$ and $X^{\prime }(L)=0$ are

For $C_t=D{C}_{xx}+aC$ with $a>0$, $C(0,t)=0$, and $C_x(L,t)=0$, every mode decays exactly when