The wellposedness and energy estimate for wave equations in domains with a space-like boundary

This paper is concerned with wave equations defined in domains of $\mathbb R^2$ with an invariable left boundary and a space-like right boundary which means the right endpoint is moving faster than the characteristic. Different from the case where the endpoint moves slower than the characteristic, this problem with ordinary boundary formulations may cause ill-posedness. In this paper, we propose a new kind of boundary condition to make systems well-posed, based on an idea of transposition. The key is to prove wellposedness and a hidden regularity for the corresponding backward system. Moreover, we establish an exponential decay estimate for the energy of homogeneous systems