Initial layer ($\sigma =t/ϵ$)

An initial layer is a short transient needed when the reduced outer solution cannot satisfy the original initial condition. On an inner time scale, terms multiplied by a small parameter remain comparable to the retained dynamics. For $ϵc_t=1-(1+\beta )c$, $c(0)=0$, the inner solution $c=(1-e^{-(1+\beta )t/ϵ})/(1+\beta )$ matches the outer value $1/(1+\beta )$ while retaining the initial datum.