Asymptotics of the solution of the Cauchy-Poisson problem in a layer of nonconstant thickness (Q1326923)

This article is devoted to the Cauchy-Poisson problem \[ \begin{alignedat}{4} \varepsilon^ 2 \Delta\Phi +\Phi_{yy} &=0 &\quad &(-H(x)< y<0), &\qquad &\Phi_ y+\varepsilon^ 2 \nabla \Phi\nabla H = 0 &\quad &(y=- H(x)),\\ \varepsilon^ 2 \Phi_{tt} +\Phi_ y &= 0 &\quad &(y=0), &\qquad &\Phi= \varphi_ 0,\quad \varepsilon\Phi_ t =\varphi_ 1 &\quad &(y=0,\;t=0).\end{alignedat} \] The main result is asymptotic representations of solutions to this problem for sufficiently small \(t\) in the neighborhood of some set of singularities that is naturally defined with solutions of the Hamiltonian system with Hamiltonian \({\mathcal H}= [| p| th(| p| H(X))]^{1/2}\).