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

Shibata, Yoshihiro; Tsutsumi, Yoshio

Nonlinear Anal., Theory Methods Appl. 11, 335-365 (1987).

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

MSC:

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

Keywords:

fully nonlinear; initial boundary value problem; nonlinear wave equation; unique local solution; function spaces; local existence; ellipticity

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