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

We present solutions of the boundary-value problem $$ U\left( {0,x} \right)={u_0},\,\,\,\,U\left( {t,0} \right)={u_1} $$ and the external boundary-value problem $$ U\left( {0,x} \right)={v_0},\,\,\,\,\,U\left( {t,x} \right){|_{\varGamma }}={v_1},\,\,\,\,\mathop{\lim}\limits_{{\left\| x \right\|H\to \infty }}U\left( {t,x} \right)={v_2} $$ for the nonlinear hyperbolic equation $$ \frac{{{\partial^2}U\left( {t,x} \right)}}{{\partial {t^2}}}+\alpha \left( {U\left( {t,x} \right)} \right){{\left[ {\frac{{\partial U\left( {t,x} \right)}}{{\partial t}}} \right]}^2}={\varDelta_L}U\left( {t,x} \right) $$ with infinite-dimensional Levy Laplacian Δ L