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

For a nonlinear hyperbolic equation with variable coefficients and the infinite-dimensional Lévy Laplacian $\Delta _L$,
$$ \beta\biggl(\sqrt{2}\mspace{2mu}\|x\|_H \frac{\partial U(t,x)}{\partial t}\biggr) \frac{\partial^2U(t,x)}{\partial t^2} +\alpha(U(t,x)) \biggl[\frac{\partial U(t,x)}{\partial t}\biggr]^2 =\Delta_LU(t,x), $$
we present algorithms for the solution of the boundary-value problem $U(0,x)=u_0$, $U(t,0)=u_1$ and the exterior boundary-value problem $U(0,x)=v_0$, $U(t,x)|_\Gamma=v_1$, $\lim_{\|x\|_H\to\infty}U(t,x)=v_2$ for the class of Shilov functions depending on the parameter $t$.