Boundary-Value Problems for a Nonlinear Hyperbolic Equation with Variable Coefficients and the Lévy Laplacian. II
M. N. Feller Ukrainian Research Institute "Resource", Kiev
Abstract:
For the following nonlinear hyperbolic equation with variable coefficients and the infinite-dimensional Lévy Laplacian
$\Delta_L$,
\begin{align*}
&\biggl(\sqrt{2}\|x\|_H \frac{\partial U(t,x)}{\partial t} \ln\frac{1}{\sqrt{2}\|x\|_H (\partial U(t,x)/\partial t)}\biggr)^{-1} \frac{\partial^2U(t,x)}{\partial t^2} -\alpha(U(t,x)) \biggl[\frac{\partial U(t,x)}{\partial t}\biggr]^2 \\
&\qquad =\Delta_LU(t,x),
\end{align*}
formulas for the solution of the boundary-value problem
$$
U(0,x)=u_0,\qquad U(t,0)=u_1
$$
and of the exterior boundary-value problem
$$
U(0,x)=v_0,\qquad U(t,x)|_\Gamma=v_1,\qquad \lim_{\|x\|_H \to\infty}U(t,x)=v_2
$$
are obtained.
Keywords:
nonlinear hyperbolic equation, Lévy Laplacian, boundary-value problem, exterior boundary-value problem, Shilov function class.
UDC:
517.9
Received: 07.08.2014
DOI:
10.4213/mzm10660
Fulltext:
PDF file (444 kB)
