Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space H the Cauchy problem (P0P0): u′(t)+Au(t)+Bu(t)=f(t)u′(t)+Au(t)+Bu(t)=f(t), 0≤t≤T0≤t≤T; u(0)=u0u(0)=u0, where −A   is the infinitesimal generator of a C0C0-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 (P0P0) the following regularization (PεPε): −εu″(t)+u′(t)+Au(t)+Bu(t)=f(t)−εu″(t)+u′(t)+Au(t)+Bu(t)=f(t), 0≤t≤T0≤t≤T; u(0)=u0u(0)=u0, u′(T)=uTu′(T)=uT, where ε>0ε>0 is a small parameter. We investigate existence, uniqueness and higher regularity for problem (PεPε). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (PεPε). Problem (PεPε) 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)C([0,T];H). However, the boundary layer of order one is not visible through the norm of L2(0,T;H)L2(0,T;H).