On an application of conservation laws in asymptotic problems for equations with operatorvalued symbol. (Russian) Zbl 0731.35062

Dobrokhotov, S. Yu.; Tolstova, O. L.

Mat. Zametki 47, No. 5, 148-151 (1990).

The following equations (elasto-dynamics) are considered \((h\ll 1)\) \[ h^ 2\nabla^ 2u_ i+u_{izz}+(c^ 2-1)h\frac{\partial}{\partial x_ i}(h\nabla u+u_{3z})-h^ 2\mu^ 2u_{itt}=0,\quad i=1,2, \] \[ h^ 2\nabla^ 2u_ 3+u_{3zz}+(c^ 2-1)\frac{\partial}{\partial z}(h\nabla u+u_{3z})-h^ 2\mu^ 2u_{3tt}=0,\quad z<-H(x); \] \[ h^ 2\nabla^ 2\phi +\phi_{zz}=0,\quad -H(x)<z<0; \] together with a system of boundary value conditions. The solution is sought in the form \[ \left( \begin{matrix} \phi \\ U\end{matrix} \right)=\exp (\frac{iS(x,t)}{h})\sigma (x,t)\left( \begin{matrix} \psi \\ V\end{matrix} \right)\exp (\int^{t}_{0}Md\tau), \] where \(U=(u_ 1,u_ 2,u_ 3)\). Explicit formulas are indicated for \(\psi\) and V. Under a normalizing assumption on the energy function, it is shown that \(M\equiv 0\).

Reviewer: C.Corduneanu (Arlington)

MSC:

35L55	Higher-order hyperbolic systems
35C05	Solutions to PDEs in closed form
35Q72	Other PDE from mechanics (MSC2000)
74B05	Classical linear elasticity