Singular Limit of Some Quasilinear Wave Equations with Damping and Restoring Terms

Abstract

A mixed problem for some hyperbolic equation with small parameter $\varepsilon$ under the presence of a restoring term $|u|^{\alpha}u$ and a reduced problem for a parabolic type are considered. Several $\varepsilon$ weighted energy estimates can be obtained by the method of difference quotients. It is shown that the solution $u_\varepsilon$ of the mixed problem converges, uniformly on any finite time interval, to the solution $u$ of the problem for the parabolic equation in an appropriate Hilbert space as $\varepsilon\to 0$.