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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
Full Text: DOI
[1] Browder, F., On the spectral theory of elliptic partial differential operators, I, Math. annaln, 142, 22-130, (1961) · Zbl 0104.07502
[2] Dafermos, C.M.; Hrusa, W.J., Energy method for quasilinear hyperbolic initial-boundary value problems. applications to elastodynamics, Archs ration. mech. analysis, 87, 267-292, (1985) · Zbl 0586.35065
[3] Dionne, P., Sur LES problḿe de Cauchy hyperboliques bien poses, J. analyse math., 10, 1-90, (1962) · Zbl 0112.32301
[4] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1977), Springer Berlin · Zbl 0691.35001
[5] Ikawa, M., Mixed problems for hyperbolic equations of second order, J. math. soc. Japan, 20, 581-604, (1969)
[6] John, F., Delayed singularities formation in solutions of nonlinear wave equations in higher dimensions, Communs pure appl. math., 29, 649-681, (1976)
[7] Kato, T., Quasilinear equations of evolution with applications to partial differential equations, (), 25-70
[8] Kato, T., Linear and quasi-linear equations of evolution of hyperbolic type, C. i. m. e., II, 127-191, (1976)
[9] Klainerman, S., Global existence for nonlinear wave equations, Communs pure appl. math., 33, 43-101, (1980) · Zbl 0405.35056
[10] Klainerman, S.; Ponce, G., Global small amplitude solutions to nonlinear evolution equations, Communs pure appl. math., 36, 133-141, (1983) · Zbl 0509.35009
[11] Matsumura, A.; Nishida, T., The initial value problem for the equations of motion of viscous and heat-conductive gases, J. math. Kyoto univ., 20, 67-104, (1980) · Zbl 0429.76040
[12] Mizohata, S., The theory of partial differential equations, (1973), Cambridge University Press London · Zbl 0263.35001
[13] Mizohata, S., Quelque probléme au bord, du type mixte, pour equations hyperboliques, (), 23-60
[14] Rabinowitz, P.H., Periodic solutions of nonlinear hyperbolic partial differential equations II, Communs pure appl. math., 22, 15-39, (1969) · Zbl 0157.17301
[15] Seeley, R.T., Extension of C∞ functions defined in a half space, Proc. am. math. soc., 15, 625-626, (1964) · Zbl 0127.28403
[16] Shatah, J., Global existence of small solutions to nonlinear evolution equations, J. diff. eqns, 46, 409-425, (1982) · Zbl 0518.35046
[17] Shibata, Y., On the global existence of classical solutions of mixed problem for some second order non-linear hyperbolic operators with dissipative term in the int erior domain, Funkcialaj ekvacioj, 25, 303-345, (1982) · Zbl 0524.35070
[18] Shibata, Y., On the global existence of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domai n, Tsukuba J. math., 7, 1-68, (1983) · Zbl 0524.35071
[19] Shibata, Y.; Tsutsumi, Y., Global existence theory of nonlinear wave equation in exterior domain, (), 155-196
[20] Shibata, Y.; Tsutsumi, Y., On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain, Math. Z., 191, 165-199, (1986) · Zbl 0592.35028
[21] Chen, V.C.; von Wahl, W., Das rand-anfangswertproblem für quasilineare wellenleichungen in sobolevräumen niedriger ordnung, J. reine angew. math., 337, 77-112, (1982) · Zbl 0486.35053
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