Time Singular Limit of Semilinear Wave Equations with Damping

The hyperbolic semilinear initial value problem ϵutt + Aut + Bu + f(u) = 0, u(0) = u0ϵ, ut(0) = u1ϵ, with commuting positive selfadjoint operators A and B in a Hilbert space X is considered. The term Aut is a damping term. It is shown that the solutions converge, uniformly in time, in an appropriate Hilbert space Z, to the solution of the parabolic type initial value problem ut + A−1Bu + A−1f(u) = 0, u(0) = u00 provided u0ϵ converge to u00, and u1ϵ, f, A, and B satisfy certain conditions. The applications include different initial-boundary value problems from continuum mechanics