On some nonlinear hyperbolic systems with damping boundary conditions.

*(English)*Zbl 0772.35031The 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)].

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)].

Reviewer: S.A.Spagnolo (Pisa)

##### Keywords:

Sobolev space; initial-boundary value problem; existence; nonlinear hyperbolic systems; nonlinear elastodynamics equation; quasilinear scalar equations
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\textit{Y. Shibata} and \textit{S. Zheng}, Nonlinear Anal., Theory Methods Appl. 17, No. 3, 233--266 (1991; Zbl 0772.35031)

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