Energy bounds for a moving boundary initial boundary value problem for the semi-linear wave equation

The context of the paper is an initial boundary value problem for the semi-linear wave equation in a straight-line region with assigned moving boundaries, upon which time-dependent Dirichlet conditions are specified. Assuming smooth solutions, bounds, in terms of data, are obtained for the energy when the boundary motions are suitably restricted, and their sharpness (in a technical sense) is established (i.e. circumstances in which the bounds become equalities are obtained). It is shown how point-wise, upper bounds for the solution follow in the case of upper bounds for the energy. The means by which a generalization of the energy bounds to a three-dimensional context may be obtained is outlined briefly.