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Asymptotics of the solution of the Cauchy-Poisson problem in a layer of nonconstant thickness. (English. Russian original) Zbl 0810.35084

Math. Notes 53, No. 6, 657-660 (1993); translation from Mat. Zametki 53, No. 6, 141-145 (1993).
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 \({\mathcal H}= [| p| th(| p| H(X))]^{1/2}\).
Reviewer: P.Zabreiko (Minsk)

MSC:

35Q35 PDEs in connection with fluid mechanics
35C20 Asymptotic expansions of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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