The boundness of the operator-valued functions for multidimensional nonlinear wave equations with applications

The operator-variation-of-constants formula was derived by Wu et al. (2015) for the general multidimensional nonlinear wave equations, and the authors proved that the formula is adapted to different boundary conditions. Furthermore, an energy-preserving scheme for one-dimensional (1D) nonlinear Hamiltonian wave equations with periodic boundary conditions was proposed by Liu et al. (2016). It is known that the formula is associated with the operator-valued functions which depend on Laplacian. Hence, it is crucial to show the boundness of the operator-valued functions. This motivates the new study of the boundness of the operator-valued functions. As an application, we extend the energy-preserving scheme from 1D to multidimensional nonlinear Hamiltonian wave equations with three different boundary conditions.