  We study the well-posedness of the Cauchy problem with Dirichlet or Neumann
boundary conditions associated to an H 1 -critical semilinear wave equation on
a smooth bounded 2D domain {\Omega}. First, we prove an appropriate Strichartz
type estimate using the $L^q$ spectral projector estimates of the Laplace
operator. Our proof follows Burq-Lebeau-Planchon [5]. Then, we show the global
well-posedness when the energy is below or at the threshold given by the sharp
Moser-Trudinger inequality. Finally, in the supercritical case, we prove an
instability result using the finite speed of propagation and a quantitative
study of the associated ODE with oscillatory data.
