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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)].

##### MSC:
 35L60 First-order nonlinear hyperbolic equations 74B20 Nonlinear elasticity
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##### References:
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