L 2 ‐convergence results for linear dissipative wave equations in unbounded domains

Abstract

Hyperbolic linear Cauchy problem εu″+Au+u′=0, u(0)=u₀, u′(0)=u₁, with “nonnegative” selfadjoint operator A in a real Hilbert space H is first considered. It is shown that the solution u_ε tends to some solution v as ε↓0 for the parabolic equation v′+Av=0 in a certain sense. Some applications are given. Finally, we present hyperbolic‐hyperbolic convergence results such as the solution for the damped wave equations goes to some solution for the free wave equations as the effect of the damping vanishes in a concrete context.