Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space $H$ the Cauchy problem
$(P_{0})\colon u^{\prime}(t)+Au(t)+Bu(t)=f(t), \, 0\leq t \leq T; \, u(0)=u_{0}$,
where $-A$ is the infinitesimal generator of a $C_0$-semigroup of contractions,
$B$ is a nonlinear monotone operator, and $f$ is a given $H$-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem $(P_{0})$ the following regularization:
$(P_{\varepsilon})\colon -\varepsilon u^{\prime \prime}(t)+u^{\prime}(t)+Au(t)+Bu(t)=f(t), \, 0\leq t \leq T; \,u(0)=u_{0}, \, u^{\prime}(T)=u_{T},$
where $\varepsilon >0$ is a small parameter. We investigate existence, uniqueness
and higher regularity for problem $(P_{\varepsilon})$. Then we establish asymptotic
expansions of order zero, and of order one, for the solution of $(P_{\varepsilon})$. Problem $(P_{\varepsilon})$ turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of $C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of $L^{2}(0,T;H)$.