On wave equations with boundary dissipation of memory type. (English) Zbl 0858.35075
Summary: The undamped wave equation on an open domain of arbitrary dimension and boundary of class \(C^1\) is considered. On parts of the boundary the normal derivative of the solution equals the convolution of its time derivative with a measure of positive type. This setting subsumes standard dissipative boundary conditions as well as the interaction with viscoelastic boundary materials. Applying methods for evolutionary integral equations to a variational formulation of the problem, existence, uniqueness and regularity of the solution to the wave equation is proven under minimal regularity assumptions on the initial conditions and forcing functions. To evaluate the versatility of a parametrized model, least-squares fits to physical data are presented.
MSC:
| 35L05 | Wave equation |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
References:
| [1] | W. Arendt, vector Laplace transforms and Cauchy problems , Israel J. Math. 59 (1987), 327-352. · Zbl 0637.44001 · doi:10.1007/BF02774144 |
| [2] | H.T. Banks, G. Propst and R.J. Silcox, A comparison of time domain boundary conditions for acoustic waves in wave guides , Quart. Appl. Math., · Zbl 0858.35074 |
| [3] | J.T. Beale, Spectral properties of an acoustic boundary condition , Indiana Univ. Math. J. 25 (1976), 895-917. · Zbl 0325.35060 · doi:10.1512/iumj.1976.25.25071 |
| [4] | G. Chen, A note on the boundary stabilization of the wave equation , SIAM J. Control Optim. 19 (1981), 106 -113. · Zbl 0461.93036 · doi:10.1137/0319008 |
| [5] | J. Diestel and J.J. Uhl, Jr., Vector measures , Math. Surveys Monographs 15 , Amer. Math. Soc., Providence, RI, 1977. · Zbl 0369.46039 |
| [6] | R. Leis, Initial boundary value problems in mathematical physics , John Wiley and Sons, New York, 1986. · Zbl 0599.35001 |
| [7] | V. Komornik, Rapid boundary stabilization of the wave equation , SIAM J. Control Optim. 29 (1991), 197-208. · Zbl 0749.35018 · doi:10.1137/0329011 |
| [8] | J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation , J. Differential Equations 50 (1983), 163-182. · Zbl 0536.35043 · doi:10.1016/0022-0396(83)90073-6 |
| [9] | A. Majda, Disappearing solutions for the dissipative wave equation , Indiana Univ. Math. J. 24 (1975), 1119-1133. · Zbl 0329.35038 · doi:10.1512/iumj.1975.24.24093 |
| [10] | ——–, The location of the spectrum for the dissipative acoustic operator , Indiana Univ. Math. J. 25 (1976), 973-987. · Zbl 0357.35064 · doi:10.1512/iumj.1976.25.25077 |
| [11] | P.M. Morse and K.U. Ingard, Theoretical acoustics , McGraw-Hill, New York, 1968. |
| [12] | NAG, Fortran library manual , Mark 14, The Numerical Algorithms Group, Ltd., Oxford, 1990. |
| [13] | J.A. Nohel and D.F. Shea, Frequency domain methods for Volterra equations , Adv. Math. 22 (1976), 278-304. · Zbl 0349.45004 · doi:10.1016/0001-8708(76)90096-7 |
| [14] | J. Prüss, Evolutionary integral equations and applications , Birkhäuser Verlag, Basel, 1993. · Zbl 0784.45006 |
| [15] | J.P. Quinn and D.L. Russell, Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping , Proc. Roy Soc. Edinburgh Sect. A 77 (1977), 97-127. · Zbl 0357.35006 |
| [16] | A. Wyler, Über die Stabilität von dissipativen Wellengleichungen , Ph.D. thesis, Univ. Zürich, 1992. · Zbl 0153.34003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
