 Heat equation with a single stationary sine forcing

On $0<x<L$, let $u_t=\nu u_{xx}+G\sin (n\pi x/L)$, with $\nu >0$, zero Dirichlet boundary conditions, and zero initial data. Write $k_n=n\pi /L$. Its stationary solution is $u_s=G\sin (k_{n}x)/(\nu k_n^2)$. Subtracting it removes the forcing from the partial differential equation and leaves a heat equation with initial data $-u_s$.

Because the forcing is a single spatial eigenfunction, separation of variables gives the entire solution without mixing modes:Its integral is $G[1-(-1{)}^n](1-e^{-\nu k_n^{2}t})/(\nu k_n^3)$; even modes have zero signed net integral. Removing forcing from the evolution equation does not remove its amplitude from the initial data for the transient.