Boundary-value problems for a nonlinear hyperbolic equation with divergent part and Levy Laplacian

Abstract

We propose an algorithm for the solution of the boundary-value problem $U(0,x) = u_0,\;\; U(t, 0) = u_1$ and the external boundary-value problem $U(0, x) = v_0, \;\;U(t, x) |_{\Gamma} = v_1, \;\; \lim_{||x||_H \rightarrow \infty} U(t, x) = v_2$ for the nonlinear hyperbolic equation $$\frac{\partial}{\partial t}\left[k(U(t,x))\frac{\partial U(t,x)}{\partial t}\right] = \Delta_L U(t,x)$$ with divergent part and infinite-dimensional Levy Laplacian $\Delta_L$.