On the Critical Semilinear Wave Equation outside Convex Obstacles
نویسندگان
چکیده
In this paper we shall show that certain estimates for the Euclidean wave equation also hold on Riemannian manifolds with smooth, strictly geodesically concave boundaries. By the last condition, we understand that the second fundamental form on the boundary of the manifold is strictly positive definite. We shall then give two applications of our estimates. First, we shall show that if n is the exterior of a smooth, compact, and strictly convex obstacle ~ c lle , then there exists a unique global, smooth solution to the critical wave equation in R+ x n:
منابع مشابه
On the Critical Semilinear Wave Equation outside Convex Obstacles Hart
In this paper we shall show that certain estimates for the Euclidean wave equation also hold on Riemannian manifolds with smooth, strictly geodesically concave boundaries. By the last condition, we understand that the second fundamental form on the boundary of the manifold is strictly positive definite. We shall then give two applications of our estimates. First, we shall show that if n is the ...
متن کاملOn the Schrodinger equation outside strictly convex obstacles
We prove sharp Strichartz estimates for the semi-classical Schrödinger equation on a compact Riemannian manifold with smooth, strictly geodesically concave boundary. We deduce classical Strichartz estimates for the Schrödinger equation outside a strictly convex obstacle, local existence for the H1-critical (quintic) Schrödinger equation and scattering for the sub-critical Schrödinger equation i...
متن کاملNull Form Estimates for (1/2,1/2) Symbols and Local Existence for a Quasilinear Dirichlet-wave Equation
We establish certain null form estimates of Klainerman-Machedon for parametrices of variable coefficient wave equations for the convex obstacle problem, and for wave equations with metrics of bounded curvature. These are then used to prove a local existence theorem for nonlinear Dirichlet-wave equations outside of convex obstacles.
متن کاملGeneralized Strichartz Estimates on Perturbed Wave Equation and Applications on Strauss Conjecture
The purpose of this paper is to show a general Strichartz estimate for certain perturbed wave equation under known local energy decay estimates, and as application, to get the Strauss conjecture for several convex obstacles in n = 3, 4. Our results improve on earlier work in Hidano, Metcalfe, Smith, Sogge and Zhou [14]. First, and most important, we can drop the nontrapping hypothesis and handl...
متن کاملHyperbolic Trapped Rays and Global Existence of Quasilinear Wave Equations
The purpose of this paper is to give a simple proof of global existence for quadratic quasilinear Dirichlet-wave equations outside of a wide class of compact obstacles in the critical case where the spatial dimension is three. Our results improve on earlier ones in Keel, Smith and Sogge [9] in several ways. First, and most important, we can drop the star-shaped hypothesis and handle non-trappin...
متن کامل