ALMOST GLOBAL EXISTENCE FOR 3-D QUASILINEAR WAVE EQUATIONS IN EXTERIOR DOMAINS WITH NEUMANN BOUNDARY CONDITIONS

We are concerned with the initial-boundary value problems of 3-D quasilinear wave equations outside compact convex obstacles with Neumann boundary conditions. When the surfaces of 3-D compact convex obstacles are smooth and the quadratic nonlinearities in the quasilinear wave equations do not fulfill the null condition, we establish the almost global existence of smooth small-amplitude solutions to the above initial-boundary value problems. The lower bound of the lifespan is proved to be e(kappa/epsilon) with some positive kappa and the small positive parameter epsilon as the size of the initial data. This lower bound is optimal as shown in the 3-D boundaryless case.

Automated summary

This article focuses on the initial-boundary value problems of 3-D quasilinear wave equations outside compact convex obstacles with Neumann boundary conditions. It establishes the almost global existence of smooth small-amplitude solutions to these problems when the surfaces of the obstacles are smooth and the quadratic nonlinearities do not fulfill the null condition. The lower bound of the lifespan is proven to be optimal, as shown in the 3-D boundaryless case.