Elliptic-like regularization of semilinear evolution equations

Abstract

Consider in a real Hilbert space the Cauchy problem (P0): u′(t)+Au(t)+Bu(t) = f (t), 0 ≤ t ≤ T ; u(0) = u_0, where −A is the generator of a C_0-semigroup of linear contractions and B is a smooth nonlinear operator. We associate with (P_0) the following problem: (Pε): −εu′′(t) + u′(t) + Au(t) + Bu(t) = f (t), 0 ≤ t ≤ T ; u(0) = u_0, u(T ) = u_1, where ε > 0 is a small parameter. Existence, uniqueness and higher regularity for both (P0) and (Pε) are investigated and an asymptotic expansion for the solution of problem (Pε) is established, showing the presence of a boundary layer near t = T .