An asymptotic expansion for some second order differential equations in Hilbert spaces (Q2711409)

The author compares the solution to the boundary value problem NEWLINE\[NEWLINE\varepsilon u''_\varepsilon- u_\varepsilon'= Au_\varepsilon+ f,\quad\text{a.e. on }[0,T],\quad u_\varepsilon(0)= a,\quad u_\varepsilon(T)= b,NEWLINE\]NEWLINE with the solution to the problem NEWLINE\[NEWLINEu'+Au=-f,\quad\text{a.e. on }[0,T],\quad u(0)= a.NEWLINE\]NEWLINE Here, \(\varepsilon\) is a small parameter, \(A\) is a linear operator on a Hilbert space \(H\), \(f\) is a given function, \(a,b\in D(A)\) are given.NEWLINENEWLINENEWLINEThe main goal is to show that the difference \(u_\varepsilon-u\) approximates zero on a subset \([0,\delta]\) for some \(\delta\), with \(0<\delta< T\). Moreover, since \(u(t)\) does not approximate \(u_\varepsilon(t)\) for small \(\varepsilon\) in the boundary layer \((\delta, T]\) the author constructs an asymptotic expansion for \(u_\varepsilon(t)\), that is valid in the whole interval \([0,T]\).