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**Decay of solutions of wave equations in a bounded region with boundary dissipation.**
*(English)*
Zbl 0536.35043

It has been proved an energy decay rate for solution of the wave equation \((1)\quad \partial^ 2w/\partial t^ 2-\Delta_ nw=0\) in \((0,\infty)\times \Omega\) with boundary conditions \((2)\quad w(x,t)=0\) on \(\Gamma_ 0\times [0,\infty)\), \((3)\quad \partial w/\partial \nu +a(x)\partial w/\partial t=0\) on \([0,\infty)\times \Gamma_ 1\). \(\Omega\) is a bounded, open, connected set in \(R^ n(n\geq 2)\) having a boundary \(\Gamma\) which is of class \(C^ 2\) and which consists of two parts \(\Gamma_ 0\) and \(\Gamma_ 1\), with \(\Gamma_ 1\neq 0\) and relatively open in \(\Gamma\). \(\Gamma_ 0\) is assumed to be either empty or to have a nonempty interior. \(\Gamma_ 0\) is a reflecting surface and \(\Gamma_ 1\) an energy absorbing surface, \(\nu\) is the unit normal of \(\Gamma\) pointing towards the exterior of \(\Omega\), \(a\in C^ 1({\bar \Gamma}_ 1)\), with \(a(x)\geq a_ 0>0\) on \(\Gamma_ 1\). The main result:

Theorem 1. Assume there is a vector field \(l(x)=(l_ 1(x),...,l_ n(x))\) of class \(C^ 2({\bar \Omega})\) such that (i) \(l,\nu \leq 0\) a.e. on \(\Gamma_ 0\), (ii) \(l\cdot \nu \geq \gamma>0\) a.e. on \(\Gamma_ 1\), (iii) (\(\partial l_ i/\partial x_ j+\partial l_ j/\partial x_ i)\) is uniformly positive definite on \({\bar \Omega}\). Then there are positive constants C,\(\delta\), such that \(E(w,t)\leq Ce^{-\delta t}E(w,0),\) \(t\geq 0\) for every solution of (1), (2), (3) for which \(E(w,0)<\infty.\) The condition \(E(w,0)<\infty\) means that \(W(\cdot,0)\in H^ 1(\Omega)\), \(w_ t(\cdot,0)\in L^ 2(\Omega)\) and \(w(x,0)=0\), on \(\Gamma_ 0\) if \(\Gamma_ 0\neq 0\).

Theorem 1. Assume there is a vector field \(l(x)=(l_ 1(x),...,l_ n(x))\) of class \(C^ 2({\bar \Omega})\) such that (i) \(l,\nu \leq 0\) a.e. on \(\Gamma_ 0\), (ii) \(l\cdot \nu \geq \gamma>0\) a.e. on \(\Gamma_ 1\), (iii) (\(\partial l_ i/\partial x_ j+\partial l_ j/\partial x_ i)\) is uniformly positive definite on \({\bar \Omega}\). Then there are positive constants C,\(\delta\), such that \(E(w,t)\leq Ce^{-\delta t}E(w,0),\) \(t\geq 0\) for every solution of (1), (2), (3) for which \(E(w,0)<\infty.\) The condition \(E(w,0)<\infty\) means that \(W(\cdot,0)\in H^ 1(\Omega)\), \(w_ t(\cdot,0)\in L^ 2(\Omega)\) and \(w(x,0)=0\), on \(\Gamma_ 0\) if \(\Gamma_ 0\neq 0\).

Reviewer: W.Sadkowski

### MSC:

35L20 | Initial-boundary value problems for second-order hyperbolic equations |

35L05 | Wave equation |

35B40 | Asymptotic behavior of solutions to PDEs |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

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### References:

[1] | Aronszajn, N, Sur l’unicité du prolongement des solutions aux derivées partielles elliptiques du second ordre, C. R. acad. sci. Paris, 242, 723-725, (1956) · Zbl 0074.31203 |

[2] | Chen, G, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. math. pures appl. (9), 58, 249-274, (1979) |

[3] | Chen, G, A note on the boundary stabilization of the wave equation, SIAM J. control optim., 19, 106-113, (1981) · Zbl 0461.93036 |

[4] | Hanna, M.S; Smith, K.T, Some remarks on the Dirichlet problem in piecewise smooth domains, Comm. pure appl. math., 20, 575-593, (1967) · Zbl 0154.13003 |

[5] | Kato, T, Perturbation theory for linear operators, (1966), Springer-Verlag New York · Zbl 0148.12601 |

[6] | Morawetz, C.S, Decay of solutions of the exterior problem for the wave equation, Comm. pure appl. math., 28, 229-264, (1975) · Zbl 0304.35064 |

[7] | Morawetz, C.S; Ralston, J.V; Strauss, W.A, Decay of solutions of the wave equation outside nontrapping obstables, Comm. pure appl. math., 30, 447-508, (1970) · Zbl 0372.35008 |

[8] | Quinn, J.P; Russell, D.L, Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping, (), 97-127 · Zbl 0357.35006 |

[9] | Rauch, J; Taylor, M.E, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana univ. math. J., 24, 79-86, (1974) · Zbl 0281.35012 |

[10] | Russell, D.L, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM rev., 20, 639-739, (1978) · Zbl 0397.93001 |

[11] | Shamir, E, Regularization of mixed second-order elliptic problems, Israel J. math., 6, 150-168, (1968) · Zbl 0157.18202 |

[12] | Slemrod, M, Stabilization of boundary control systems, J. differential equations, 22, 402-415, (1976) · Zbl 0304.93022 |

[13] | Strauss, W.A, Dispersal of waves vanishing on the boundary of an exterior domain, Comm. pure appl. math., 28, 265-278, (1975) · Zbl 0297.35047 |

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