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

Abstract Consider in a Hilbert space H the Cauchy problem ( P 0 ) : u ′ ( t ) + A u ( t ) + B u ( t ) = f ( t ) , 0 ≤ t ≤ T ; u ( 0 ) = u 0 , where A : D ( A ) ⊂ H → H is the generator of a C 0 -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 ( P 0 ) the approximate problem ( P e ) : − e u ″ ( t ) + u ′ ( t ) + A u ( t ) + B u ( t ) = f ( t ) , 0 ≤ t ≤ T ; u ( 0 ) = u 0 , u ( T ) = u T , where e is a positive small parameter. We establish an asymptotic expansion of the solution u e of ( P e ) , showing that u e corrected by a boundary layer function approximates the solution of ( P 0 ) 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. Morosanu [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.