Boundary-value problems for a nonlinear hyperbolic equation with variable coefficients and the Lévy Laplacian. II

Abstract

For the following nonlinear hyperbolic equation with variable coefficients and the infinite-dimensional Lévy Laplacian Δ _L ,

$\begin{gathered} \left( {\sqrt 2 \left\| x \right\|_H \frac{{\partial U(t,x)}} {{\partial t}}\ln \frac{1} {{\sqrt 2 \left\| x \right\|_H (\partial U(t,x)/\partial t)}}} \right)^{ - 1} \frac{{\partial ^2 U(t,x)}} {{\partial t^2 }} - \alpha (U(t,x))\left[ {\frac{{\partial U(t,x)}} {{\partial t}}} \right]^2 \hfill \\ = \Delta _L U(t,x), \hfill \\ \end{gathered} $

formulas for the solution of the boundary-value problem

$U(0,x) = u_0 , U(t,0) = u_1 $

and of the exterior boundary-value problem

$U(0,x) = v_0 , U(t,x)|_\Gamma = v_1 \mathop {\lim }\limits_{\left\| x \right\|_H \to \infty } U(t,x) = v_2 $

are obtained.