  In this paper, the Cauchy problem for linear and nonlinear convolution wave
equations are studied.The equation involves convolution terms with a general
kernel functions whose Fourier transform are operator functions defined in a
Banach space E together with some growth conditions. Here, assuming enough
smoothness on the initial data and the operator functions, the local, global
existence, uniqueness and regularity properties of solutions are established in
terms of fractional powers of given sectorial operator functon. Furthermore,
conditions for finite time blow-up are provided. By choosing the space E and
the operators, the regularity properties the wide class of nonlocal wave
equations in the field of physics are obtained.
