Boundary-value problems for a nonlinear hyperbolic equation with Lévy Laplacian

We present solutions of the boundary-value problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ U\left( {0,x} \right)={u_0},\,\,\,\,U\left( {t,0} \right)={u_1} $$\end{document}and the external boundary-value problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ U\left( {0,x} \right)={v_0},\,\,\,\,\,U\left( {t,x} \right){|_{\varGamma }}={v_1},\,\,\,\,\mathop{\lim}\limits_{{\left\| x \right\|H\to \infty }}U\left( {t,x} \right)={v_2} $$\end{document}for the nonlinear hyperbolic equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{{{\partial^2}U\left( {t,x} \right)}}{{\partial {t^2}}}+\alpha \left( {U\left( {t,x} \right)} \right){{\left[ {\frac{{\partial U\left( {t,x} \right)}}{{\partial t}}} \right]}^2}={\varDelta_L}U\left( {t,x} \right) $$\end{document}with infinite-dimensional Lévy Laplacian ΔL: