The Cauchy problem for a semilinear wave equation. III

The Cauchy problem for a semilinear pseudodifferential second order hyperbolic equation of the form
$$ \frac{\partial^2}{\partial t^2}u(t,x)+iB(t)\frac\partial{\partial t}u(t,x)+A(t)u(t,x)+f(t,x;u(t,x))=0 $$
is studied. The results (presented in a previous author's paper, see Zapisky Nauch. Semin. LOMI, 1990, v. 182) on the existence and uniqueness of the global weak (energy class) solutions are revised. In the case of more regular initial data ($u(0,\cdot)\in H^{s+1}$, $\partial_t u(0,\cdot)\in H^s$, $0<s\leqslant2$) the respective regularity of weak solutions is proved.