Singular limit of some quasilinear wave equations with damping terms

Abstract

We consider a relation between a mixed problem for a class of quasilinear wave equations with small parameter $\epsilon$ and a reduced problem of a parabolic type. By constructing the stable set the global existence of solutions can be discussed. It is shown that the solution $u_{\epsilon}$ of the mixed problem converges, uniformly on any finite time interval, to the solution $u$ of the parabolic equation in an appropriate Hilbert space as $\epsilon \rightarrow 0$. Several $\epsilon$ weighted energy estimates will be obtained in order to evaluate the difference norm of $u_{\epsilon}-u$