Elliptic-like regularization of semilinear evolution equations and applications to some hyperbolic problems

Abstract

Consider in a Hilbert space H the Cauchy problem (P0): u′(t)+Au(t)+Bu(t)=f(t), 0≤t≤T; u(0)=u0, where A:D(A)⊂H→H is the generator of a C0-semigroup of contractions and B:H→H is Lipschitzian on bounded sets and monotone. Following the method of artificial viscosity introduced by J.L. Lions, we associate with (P0) the approximate problem (Pε): −εu″(t)+u′(t)+Au(t)+Bu(t)=f(t), 0≤t≤T; u(0)=u0, u(T)=uT, where ε is a positive small parameter. We establish an asymptotic expansion of the solution uε of (Pε), showing that uε corrected by a boundary layer function approximates the solution of (P0) with respect to the sup norm of C([0,T];H). The same asymptotic expansion still holds if B is not necessarily monotone but is Lipschitzian on H. This paper is a significant extension of a previous one by M. Ahsan and G. Moroşanu [2] so that the framework created here allows the treatment of hyperbolic problems (besides parabolic ones). Specifically, our main result is illustrated with the semilinear telegraph system (thus extending a result by N.C. Apreutesei and B. Djafari Rouhani [3]) and the semilinear wave equation.