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On a local existence theorem of Neumann problem for some quasilinear hyperbolic system of 2nd order. (English) Zbl 0652.35077
It is studied the local existence in time of classical solutions to the mixed problem for the quasilinear hyperbolic system of second order with Neumann type boundary condition. The equation is quasilinear but the boundary condition is fully nonlinear. The full nonlinearity has kept away from proving the local existence theorem, because it breaks down the usual iteration process. Our idea of solving the problem is to introduce the time derivative of the original unknown function as the new unknown function and to reduce the original problem to some hyperbolic-elliptic system.
As an application, we can show the local existence in time of classical solutions to Neumann or third kind problems for nonlinear acoustical equations. And also, as another application, we can get a local existence theorem for nonlinear elastodynamical systems with traction boundary condition when the pressure and the displacement is very small in the initial state.
Reviewer: Y.Shibata

MSC:
35L70 Second-order nonlinear hyperbolic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
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