In this paper, the integral problem for linear and nonlinear wave equations is studied. The equation involves elliptic operator L and abstract operator A in Hilbert space H. Here, assuming enough smoothness on the initial data in terms of interpolation spaces, integral conditions, and assumptions on operators A, L, the existence and uniqueness of local and global solutions and the L^p-regularity properties of solutions are established. By choosing the space H and operators L and A, the regularity properties of solutions to different classes of wave equations in the field of physics are obtained.