Elliptic-like regularization of semilinear evolution equations

Consider in a real Hilbert space the Cauchy problem (P0P0): u′(t)+Au(t)+Bu(t)=f(t),0≤t≤T;u(0)=u0, where −A−A is the generator of a C0C0-semigroup of linear contractions and BB is a smooth nonlinear operator. We associate with (P0P0) the following problem: (P1ε): −εu″(t)+u′(t)+Au(t)+Bu(t)=f(t),0≤t≤T;u(0)=u0,u(T)=u1, where ε>0ε>0 is a small parameter. Existence, uniqueness and higher regularity for both (P0P0) and (P1ε) are investigated and an asymptotic expansion for the solution of problem (P1ε) is established, showing the presence of a boundary layer near t=Tt=T.