Boundary-Value Problems for a Nonlinear Hyperbolic Equation with Variable Coefficients and the Lévy Laplacian. II

For the following nonlinear hyperbolic equation with variable coefficients and the infinite-dimensional Lévy Laplacian $\Delta_L$,
\begin{align*} &\biggl(\sqrt{2}\|x\|_H \frac{\partial U(t,x)}{\partial t} \ln\frac{1}{\sqrt{2}\|x\|_H (\partial U(t,x)/\partial t)}\biggr)^{-1} \frac{\partial^2U(t,x)}{\partial t^2} -\alpha(U(t,x)) \biggl[\frac{\partial U(t,x)}{\partial t}\biggr]^2 \\ &\qquad =\Delta_LU(t,x), \end{align*}
formulas for the solution of the boundary-value problem
$$ U(0,x)=u_0,\qquad U(t,0)=u_1 $$
and of the exterior boundary-value problem
$$ U(0,x)=v_0,\qquad U(t,x)|_\Gamma=v_1,\qquad \lim_{\|x\|_H \to\infty}U(t,x)=v_2 $$
are obtained.