Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture. (Q415311)

The paper deals with the initial value problem to the equation \(u_{tt}-\Delta _gu=F(t,x)\) in \((0,\infty )\times \Omega \) with the boundary conditions \(u(t,x)=0\) or \(\frac {\partial u}{\partial n}(t,x)=0\), \(t>0\), \(x\in \partial \Omega \), where \(\Delta _g\) is the Laplace-Beltrami operator, \(\Omega \) is either \(\mathbb {R}^n\) or the complement of a ball in \(\mathbb {R}^n\). Using the interpolation method the author proves a general Strichartz estimate for the problem under known local energy estimates. As applications, the Strauss conjecture for several convex obstacles for \(n=3,4\) is deduced.