Abstract
We consider the mixed problems for the wave equation with general boundary condition. First we discuss on the well posedness of the problems for a boundary operator with real valued coefficients and we show the necessary and sufficient condition for the well posedness in the sense of Cā when the domain is the exterior of a strictly convex object. As a consequence of the considerations on the well posedness we like to show the decay of the solutions on some additional conditions on boundary operators.
Second, we consider the decay of solutions in the exterior of convex obstacles of finite number.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Asakura F., On the Greenās function ā-Ī»2 with the boundary condition of the third kind in the exterior domain of a bounded obstacle, J.Math.Kyoto Univ., 12 (1978), 615ā625.
Eskin G., Seminaire Goulaouic-Schwartz, 1979ā1980.
Ikawa M., Mixed problem for the wave equation with an oblique derivative boundary condition, Osaka J.Math. 7 (1970) 495ā525.
Ikawa M, Remarques sur les problĆØmes mixtes pour lāequation des ondes, Colloque international du C.N.R.S., AstĆ©risque 2 et 3, 217ā221.
Ikawa M, ProblĆØmes mixtes pas nĆ©cessairement L2 -bien poses pour les Ć©quations strictement hyperboliques, Osaka J.Math., 12 (1975), 69ā115.
Ikawa M, Sur les problĆØmes mixtes pour lāĆ©quation des ondes, Puhl.Res.Inst.Math.Sci.Kyoto Univ., 10 (1975), 669ā690.
Ikawa M, ProblĆØmes mixtes pour lāĆ©quation des ondes,Publ. Res.Inst.Math.Sci.Kyoto Univ., 12 (1976), 55ā122.
Ikawa M, ProblĆØmes mixtes pour lāĆ©quation des ondes. II, Publ.Res.Inst.Math.Sci.Kyoto Univ., 13 (1977), 61ā106.
Ikawa M, Mixed problems for the wave equation. III, Exponential decay of solutions, Publ.Res.Inst.Math.Sci. Kyoto Univ., 14 (1978), 71ā110.
Ikawa M, Mixed problems for the wave equation. IV, J.Math. Kyoto Univ., 19 (1979), 375ā411.
Ikawa M, On the mixed problems for the wave equation in an interior domain.II, Osaka J.Math., 17 (1980), 253ā279.
Ikawa M, Decay of solutions of the wave equation in the exterior of two convex obstacles, to appear in Osaka J.Math.
Kajitani K., A necessary condition for the well posed hyperbolic mixed problem with variable coefficients, J.Math. Kyoto Univ., 14 (1974), 231ā242.
Lax P.D. and Phillips R.S., Scattering theory, Academic press, New York, (1967).
Keller J.B.Lewis R.M. and Seckler B.D., Asymptotic solution of some diffraction problems, Comm.Pure Appl.Math., 9 (1956), 207ā265.
Ludwig D., Uniform asymptotic expansion of the fiels scattered by a convex object at high frequencies, Comm.Pure Appl.Math., 20 (1967), 103ā138.
Miyatake S., Mixed problems for hyperbolic equations of second order with first order complex boundary operators, Japan J.Math., 1 (1975), 111ā158.
Miyatake S, A sharp form of the existence theorem for hyperbolic mixed problems of second order, J.Math.Kyoto Univ. 17 (1977), 199ā223.
Morawetz C.S., Decay for solutions of the exterior problems for the wave equation, Comm.Pure Appl.Math., 28 (1975), 229ā264.
Morawetz C.S., Ralston J. and Strauss W.A., Decay of solutions of the wave equation outside nontrapping obstacles, Comm.Pure Appl.Math., 30 (1977), 447ā508.
Ralston J., Solutions of the wave equation with localized energy, Comm.Pure Appl.Math., 22 (1969), 807ā823.
Soga H., Mixed problems in a quater space for the wave equation with a singular oblique derivative, Publ.Res.Inst. Kyoto Univ., 15 (1979), 357ā399.
Soga H., Mixed problems for the wave equation with a singular oblique derivative, Osaka J.Math., 17 (1980), 199ā232.
Walker H.F., Some remarks on the local energy decay of solutions of the initial-boundary value problem for the wave equation in unbounded domains, J.Diff.Equ., 23 (1977), 459ā471.
Kohigashi, N., On the strong hyperbolicity of mixed problems with constant coefficients in a quarter space, (in Japanese) Masterās thesis, Osaka Univ., 1979.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
Ā© 1981 D. Reidel Publishing Company
About this paper
Cite this paper
Ikawa, M. (1981). Mixed Problems for the Wave Equation. In: Garnir, H.G. (eds) Singularities in Boundary Value Problems. NATO Advanced Study Institutes Series, vol 65. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8434-9_5
Download citation
DOI: https://doi.org/10.1007/978-94-009-8434-9_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-8436-3
Online ISBN: 978-94-009-8434-9
eBook Packages: Springer Book Archive