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Global smooth solutions for the quasilinear wave equation with boundary dissipation. (English) Zbl 1214.35037

Let \(\Omega\) be a bounded open subset of a Euclidean space \(\mathbb R^n\) and \(a(x,y)\) be a smooth vector valued mapping defined on \(\overline \Omega\times\mathbb R^n\). The author considers the existence of global solutions of the quasilinear wave equation
\[ \partial^2 u/\partial t^2= \operatorname{div}a(x, \nabla u) \quad (0\leq t\leq T;\;x\in \Omega) \]
with a boundary dissipation structure of an input-output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system. The main tool in the paper is the geometrical analysis.
The main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system. It is shown that the existence of global solutions depends not only on this dissipation structure but also on a Riemannian metric, given by the coefficients and the equilibrium of the system. Some geometrical conditions on this Riemannian metric are presented to guarantee the existence of global solutions. In particular, it is proved that the norm of the state of the system decays exponentially if the input stops after a finite time, which implies the exponential stabilization of the system by boundary feedback.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B35 Stability in context of PDEs
35L65 Hyperbolic conservation laws
58J45 Hyperbolic equations on manifolds
35L20 Initial-boundary value problems for second-order hyperbolic equations
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