Energy decay for solutions of the wave equation with general memory boundary conditions

Pierre Cornilleau

Description

Energy decay for solutions of the wave equation with general memory boundary conditions. Pierre Cornilleau Ecole Centrale de Lyon Institut Camille Jordan, UMR CNRS 5208 36 avenue Guy de Collongue 69134 Ecully Cedex France Serge Nicaise Universite de Valenciennes et du Hainaut Cambresis LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes F-59313 - Valenciennes Cedex 9 France February 11, 2009 Abstract We consider the wave equation in a smooth domain subject to Dirichlet boundary conditions on one part of the boundary and dissipative boundary conditions of memory-delay type on the remainder part of the boundary, where a general borelian measure is involved. Under quite weak assumptions on this measure, using the multiplier method and a standard integral inequality we show the exponential stability of the system. Some examples of measures satisfying our hypotheses are given, recovering and extending some of the results from the literature. Introduction We consider the wave equation subject to Dirichlet boundary conditions on one part of the boundary and dissipative boundary conditions of memory-delay type on the remainder part of the boundary. More precisely, let ? be a bounded open connected set of Rn(n ≥ 2) such that, in the sense of Necˇas ([8]), its boundary ∂? is of class C2. Throughout the paper, I denotes the n?n identity matrix, while As denotes the symmetric part of a matrix A.

• u0 u?


• positive borelian


• proof ends


• measure


• inequality consequently give


• well posedness


• consequently


• dirichlet boundary


• suppose u0 ?

Sujets

• Université
• Cornilleau
• Jordan
• École centrale de Lyon
• Saint Nicaise
• Measure

Informations

• Publié par: pefav
• Nombre de lectures: 22
• Langue: English

Extrait

Energy decay for solutions of the wave equation with general memory boundary conditions. Pierre Cornilleau Ecole Centrale de Lyon Institut Camille Jordan, UMR CNRS 5208 36 avenue Guy de Collongue 69134 Ecully Cedex France [e-mail] Serge Nicaise Universite´deValenciennesetduHainautCambre´sis LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes F-59313 - Valenciennes Cedex 9 France [e-mail] February 11, 2009

Abstract We consider the wave equation in a smooth domain subject to Dirichlet boundary conditions on one part of the boundary and dissipative boundary conditions of memory-delay type on the remainder part of the boundary, where a general borelian measure is involved. Under quite weak assumptions on this measure, using the multiplier method and a standard integral inequality we show the exponential stability of the system. Some examples of measures satisfying our hypotheses are given, recovering and extending some of the results from the literature. Introduction We consider the wave equation subject to Dirichlet boundary conditions on one part of the boundary and dissipative boundary conditions of memory-delay type on the remainder part of the boundary. More precisely, let Ω be a bounded open connected set of R n ( n ≥ 2)suchthat,inthesenseofNecˇas([8]),its boundary ∂ Ω is of class C 2 . Throughout the paper, I denotes the n × n identity matrix, while A s denotes the symmetric part of a matrix A . Let m be a C 1 vectorfieldon ¯ Ωsuchthat in ¯ f div( m ) > sup(div( m ) − 2 λ m ) (1) Ω ¯ Ω where λ m ( x ) is the smallest eigenvalue function of the real symmetric matrix ∇ m ( x ) s . ll C 1 vector fields o  ¯  lds is an open cone. If m is in this set, Remark 1 The set of a n Ω such that (1) ho we denote ( m )= 2 1  Ω ¯ Ω (div( m ) − 2 λ m )   c in ¯ f div( m ) − sup Example 1 • An affine example is given by m ( x ) = ( A 1 + A 2 )( x − x 0 )  where A 1 is a definite positive matrix, A 2 a skew-symmetric matrix and x 0 any point in R n . 1