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On some nonlinear hyperbolic systems with damping boundary conditions. (English) Zbl 0772.35031
The authors consider the initial-boundary value problem with a dissipative term in the boundary condition, and prove the global existence, for small data, for a wide class of nonlinear hyperbolic systems, including the nonlinear elastodynamics equation \[ u_{tt}- \sum\partial_ i(a_ i(\varepsilon(u)))=f(t,x)\quad \text{in }\Omega,\quad\sum\nu_ i\cdot a_ i(\varepsilon(u))+b(x,u_ t)=g(t,x)\quad \text{on }\partial\Omega \] with initial conditions \(u=u_ 0\), \(u_ t=u_ 1\) at \(t=0\). Here, \(\Omega\subset\mathbb{R}^ n\) is a bounded domain with exterior normal \((\nu_ 1,\dots,\nu_ n)\), \(u(t,x)\), \(f(t,x)\), \(g(t,x)\), \(u_ 0(x)\), \(u_ 1(x)\), \(a_ i(\eta)\), \(b_ i(x,\xi)\) are \(\mathbb{R}^ n\)-valued functions in their arguments \(t>0\), \(x\in\Omega\), \(\eta\) an \(n\times n\) matrix, \(\xi\in\mathbb{R}^ n\), and \(\varepsilon(u)=[{1\over 2}(\partial_ ju_ q-\partial_ qu_ j)]\) is the strain tensor. The four-tensor \(\partial a/\partial\eta\), where \(a=(a_ 1,\dots,a_ n)\), is assumed to fulfill the usual ellipticity conditions in the theory of elasticity, while the damping matrix \([\partial b_ i/\partial\xi_ j]\) is symmetric and coercive. Moreover \(a_ i(0)=b_ i(x,0)=0\) and the data \((u_ 0,u_ 1,f,g)\) satisfy the natural compatibility conditions. Under these assumptions, there exists a solution \(u(t,x)\), for all \(t\geq 0\) and \(x\in\mathbb{R}^ n\), provided the data are sufficiently small in suitable Sobolev norms.
The same result holds true also for the quasilinear scalar equations \[ u_{tt}-\sum\partial_ i(a_ i(\partial_ 1u,\dots,\partial_ nu))=f(t,x)\;\text{in }\Omega,\quad\sum\nu_ i\cdot a_ i(\partial_ 1u,\dots,\partial_ n u)+b(x,u_ t)=g(t,x)\;\text{on }\partial\Omega \] under the coercivity assumption on the matrix \([\partial a_ i/\partial(\partial_ ju)]\) and the function \(\partial b/\partial(u_ t)\). These results are obtained after deriving a suitable decay, as \(t\to\infty\), for the solutions to the linearized equations, which in turn follows from a spectral analysis of the corresponding stationary problems. [For another paper on the same subject, see also: Tiehu Qin, Chin. Ann. Math., Ser. B 9, No. 3, 251-269 (1988; Zbl 0664.35059)].

35L60 First-order nonlinear hyperbolic equations
74B20 Nonlinear elasticity
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[1] Chen, G., Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. math. pures appl., 58, 249-273, (1976) · Zbl 0414.35044
[2] Duvaut, G.; Lions, J.L., LES inéquations en mécanique et en physique, (1972), Dunot Paris · Zbl 0298.73001
[3] Greenberg, J.M.; Li, Ta-Tsien, The effect of boundary damping for the quasilinear wave equation, J. diff. eqns, 52, 66-75, (1984) · Zbl 0576.35080
[4] Harazov, D.F., On the spectrum of completely continuous operators depending analytically on a parameter, in topological linear spaces, Acta sci. math. Szeged, 23, 38-45, (1962) · Zbl 0108.11404
[5] Ikawa, M., A mixed problem for hyperbolic equations of second order with nonhomogeneous Neumann type boundary condition, Osaka J. math., 6, 339-374, (1969) · Zbl 0207.10003
[6] Lagnese, J., Boundary stabilization of linear elastodynamics systems, SIAM J. control optim., 21, 968-984, (1983) · Zbl 0531.93044
[7] Melrose, R.B., Singularities and energy decay in acoustical scattering, Duke math. J., 46, 43-59, (1979) · Zbl 0415.35050
[8] Mizohata, S., The theory of partial differential equations, (1973), Cambridge University Press U.K · Zbl 0263.35001
[9] Nagasawa, T., On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary, J. diff. eqns, 65, 49-67, (1986) · Zbl 0598.34021
[10] Qin, Tienhu, The global smooth solutions of second order quasilinear hyperbolic equations with dissipation boundary condition, Chinese annls math., 9B, 251-269, (1988)
[11] Quinn, J.P.; Russell, D.L., Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping, Proc. R. soc. Edinburgh, 77A, 97-127, (1977) · Zbl 0357.35006
[12] Racke, R.; Shibata, Y., Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermo-elasticity, (1990), Bonn University, SFB 256
[13] Seeley, R.T., Integral equations depending analytically on a parameter, Indag. math., 24, 434-442, (1962) · Zbl 0106.08102
[14] Shen, Weixi; Zheng, Songmu, Global smooth solutions to the system of one-dimensional thermoelasticity with dissipation boundary conditions, Chinese annls math., 7B, 303-317, (1986) · Zbl 0608.73013
[15] Shibata, Y., On the global existence of classical solutions of mixed problem for some second order nonlinear hyperbolic operators with dissipative term in the inte rior domain, Funkcialaj ekvacioj, 25, 303-345, (1982) · Zbl 0524.35070
[16] Shibata, Y., On a global existence theorem of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domain, Tsukuba J. math., 7, 1-68, (1983) · Zbl 0524.35071
[17] Shibata, Y., On a local existence theorem of Neumann problem for some quasilinear hyperbolic equations, (), 133-167
[18] Shibata, Y., On the Neumann problem for some linear hyperbolic systems of second order, Tsukuba J. math., 12, 149-209, (1988) · Zbl 0674.35056
[19] Shibata, Y., On the Neumann problem for some linear hyperbolic systems of 2nd order with coefficient in Sobolev spaces, Tsukuba J. math., 13, 283-352, (1989) · Zbl 0706.35082
[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] Shibata, Y.; Nakamura, G., On a local existence theorem of Neumann problem for some quasilinear hyperbolic systems of 2nd order, Math. Z., 202, 1-64, (1989) · Zbl 0652.35077
[22] Shibata, Y.; Kikuchi, M., On the mixed problem for some quasilinear hyperbolic system with fully nonlinear boundary condition, J. diff. eqns, 80, 154-197, (1989) · Zbl 0689.35055
[23] Vainberg, B.R.; Vainberg, B.R., On the analytical properties of the resolvent for a certain class of operator-pencils, Mat. sb., Math. USSR sb., 6, 241-273, (1969)
[24] Vainberg, B.R.; Vainberg, B.R., On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as t → ∞ of solutions of nonstationary problems, Usp. mat. nauk., Russian math. survey, 30, 1-58, (1975) · Zbl 0318.35006
[25] Yamamoto, K., Theorems on singularities of solutions to systems of differential equations, Japan J. math., 14, 119-163, (1988)
[26] Zajaczkowski, W., On global existence for nonlinear wave equation in a bounded domain with dissipative boundary condition, (), SFB 256 Report, No. 15
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