Sharp Dispersive and Strichartz Estimates for the Neumann Cylindrical Wave Model

We establish local-in-time dispersive and sharp Strichartz estimates for the linear wave equation with Neumann boundary conditions inside a cylindrical convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ featuring a non-empty smooth boundary $\mathrm{\partial }\mathrm{\Omega }$. The cylindrical geometry dictates that the nonnegative radius of curvature vanishes along the axial direction. By utilizing an explicit spectral analysis based on the zeros of the Airy function derivative, alongside a tailored Airy–Poisson summation formula, we capture the microlocal behavior of multi-reflected waves and caustics. Finally, we apply these sharp Strichartz estimates without loss of derivatives to prove local well-posedness and small-data global well-posedness for the energy-critical quintic nonlinear wave equation (NLW) subject to Neumann boundary conditions.