Local existence of solution for the initial boundary value problem of fully nonlinear wave equation. (English) Zbl 0651.35053

The authors consider the following initial boundary value problem of fully nonlinear wave equation \[ (1.1)\quad Lu+F(t,x,\bar D^ 2u)=f(t,x)\quad in\quad [t_ 0,t_ 0+T]\times \Omega, \]
\[ (1.2)\quad u=0\quad on\quad (t_ 0,t_ 0+T)\times \partial \Omega, \]
\[ (1.3)\quad u(t_ 0,x)=\psi_ 0(x),\quad (\partial_ tu)(t_ 0,x)=\psi_ 1(x)\quad in\quad \Omega, \] where \[ (1.4)\quad Lu=\partial^ 2_ tu+a_ 1(t,x,\bar D^ 1_ x)\partial_ tu+a_ 2(t,x,\bar D^ 2_ x)u, \]
\[ (1.5)\quad a_ 1(t,x,\bar D^ 1_ x)u=\sum^{n}_{j=1}a_ 2^{\partial}(t,x)\partial_ ju+a^ 0_ 1(t,x)u, \]
\[ (1.6)\quad a_ 2(t,x,\bar D^ 2_ x)u=- \sum^{n}_{i,j=1}a_ 2^{i,j}(t,x)\partial_ i\partial_ ju+\sum^{n}_{j=1}a_ 1^ j(t,x)\partial_ ju+a_ 0(t,x)u, \] and \(a_ 2(t,x,\bar D^ 2_ x)\) is a strictly elliptic operator and F(t,x,\(\lambda)\) satisfies some restrictions with respect to \(a_ 2(t,x,\lambda)\) and some others.
Under some assumtions on \(\psi_ 0,\psi_ 1\), \(f,a_ 1,a_ 2\), and F the authors obtain that (1.1)-(1.3) has a unique local solution in some function spaces.
The method of proof of local existence is essentially based on ellipticity of the differential operator \(a_ 2(t,x,D^ 2_ x)\).
Reviewer: J.Wang


35L70 Second-order nonlinear hyperbolic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35L15 Initial value problems for second-order hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
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