The lifespan for general quasilinear wave equation with localized dissipation in exterior domains

Abstract

In this article, we develop the result of J. Metcalfe and M. Nakamura (General quasi-linear wave equations with localized dissipation in extension domains, J. Differ. Eqns. 233 (2007), pp. 313–344.) into a more general form. By means of the method of commuting vector fields, we get the lifespan for the quasi-linear wave equation with localized dissipations and the non-linear term depends on the unknown function itself in exterior domains.

Keywords:

• quasi-linear wave equations
• local dissipation
• obstacle
• commuting vector
• life-span

Notes

Note

1. We also note that our techniques would allow us to handle multiple-speed systems of wave equations with similar dissipations. In this article, we just prove the result in the single-speed condition for simplicity.