19 Pages
English

Stabilization of the wave equation with boundary or internal distributed delay

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Niveau: Supérieur, Doctorat, Bac+8
Stabilization of the wave equation with boundary or internal distributed delay Serge Nicaise Universite de Valenciennes et du Hainaut Cambresis LAMAV and FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes 59313 Valenciennes Cedex 9 France Cristina Pignotti Dipartimento di Matematica Pura e Applicata Universita di L'Aquila Via Vetoio, Loc. Coppito, 67010 L'Aquila Italy Abstract We consider the wave equation in a bounded region with a smooth boundary with dis- tributed delay on the boundary or into the domain. In both cases, under suitable assumptions, we prove the exponential stability of the solution. These results are obtained by introducing suitable energies and by proving some observability inequalities. For an internal distributed delay, we further show some instability results. 1 Introduction We study the wave equation subject to Dirichlet boundary conditions on one part of the boundary and dissipative boundary conditions of delay type on the remainder part of the bound- ary. More precisely, let ? ? IRn be an open bounded domain with a smooth boundary ?. We assume that ? is divided into two closed and disjoint parts ?0 and ?1, i.e. ? = ?0 ? ?1 and ?0 ? ?1 = ?. Moreover we assume that the measure of ?0 is positive. In this domain ?, we consider the initial boundary value problem utt ?∆u = 0 in ?? (0,+∞), (1.1) u = 0 on ?0 ? (0,+∞), (1.2) ∂u ∂? (t) + ∫ ?2 ?1 µ(s)ut(t?

  • initial boundary

  • parts ?0

  • product between

  • feedback

  • datum u0 ?

  • see also

  • µ1 exponential

  • ?0

  • exponential stability results

  • stability result


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Language English
Stabilization of the wave equation with boundary or internal distributed delay Serge Nicaise UniversitedeValenciennesetduHainautCambresis LAMAV and FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes 59313 Valenciennes Cedex 9 France Cristina Pignotti Dipartimento di Matematica Pura e Applicata UniversitadiLAquila Via Vetoio, Loc. Coppito, 67010 L’Aquila Italy
Abstract We consider the wave equation in a bounded region with a smooth boundary with dis-tributed delay on the boundary or into the domain. In both cases, under suitable assumptions, we prove the exponential stability of the solution. These results are obtained by introducing suitable energies and by proving some observability inequalities. For an internal distributed delay, we further show some instability results.
1 Introduction
We study the wave equation subject to Dirichlet boundary conditions on one part of the boundary and dissipative boundary conditions of delay type on the remainder part of the bound-ary. More precisely, let  IR n be an open bounded domain with a smooth boundary . We assume that  is divided into two closed and disjoint parts  0 and  1 , i.e.  =  0 1 and 0 1 = . Moreover we assume that the measure of  0 is positive. In this domain , we consider the initial boundary value problem
u tt u = 0 in  (0 , + ) , (1.1) u = 0 on  0 (0 , + ) , (1.2) u( t ) + Z 1 2 ( s ) u t ( t s ) ds + 0 u t ( t ) = 0 on  1 (0 , + ) , (1.3) u ( x, 0) = u 0 ( x ) and u t ( x, 0) = u 1 ( x ) in  , (1.4) u t ( x, t ) = f 0 ( x, t ) in  1 (0 ,  2 ) , (1.5) where ( x ) denotes the outer unit normal vector to the point x  and uis the normal derivative. Moreover, 1 and 2 are two real numbers with 0 1 <  2 ,  0 is a positive constant, 1