×

zbMATH — the first resource for mathematics

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)].

MSC:
35L60 First-order nonlinear hyperbolic equations
74B20 Nonlinear elasticity
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.