Abstract
This paper is devoted to the investigation of propagation of singularities in hyperbolic equations with non-smooth coefficients, using the Colombeau theory of generalized functions. As a model problem, we study the Cauchy problem for the one-dimensional wave equation with a discontinuous coefficient depending on time. After demonstrating the existence and uniqueness of generalized solutions in the sense of Colombeau to the problem, we investigate the phenomenon of propagation of singularities, arising from delta function initial data, for the case of a piecewise constant coefficient. We also provide an analysis of the interplay between singularity strength and propagation effects. Finally, we show that in case the initial data are distributions, the Colombeau solution to the model problem is associated with the piecewise distributional solution of the corresponding transmission problem.
Mathematics Subject Classification (2010). 35L05, 35A21, 46F30.
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References
H.A. Biagioni, A Nonlinear Theory of Generalized Functions, Lect. Notes Math. 1421, Springer-Verlag, Berlin, 1990.
J.-F. Colombeau, New Generalized Functions and Multiplication of Distributions, North-Holland Math. Stud. 84 North-Holland, Amsterdam, 1984.
J.-F. Colombeau, Elementary Introduction to New Generalized Functions, North- Holland Math. Stud. 113, North-Holland, Amsterdam, 1985.
J.-F. Colombeau and A. Heibig, Generalized solutions to Cauchy problems, Monatsh. Math. 117 (1994), 33–49.
H. Deguchi, A linear first-order hyperbolic equation with a discontinuous coefficient: distributional shadows and propagation of singularities, Electron. J. Differential Equations 201176 (2011), 1–25.
C. Garetto, Microlocal analysis in the dual of a Colombeau algebra: generalized wave front sets and noncharacteristic regularity, New York J. Math. 12 (2006), 275–318.
C. Garetto, Generalized Fourier integral operators on spaces of Colombeau type, in New Developments in Pseudo-Differential Operators, editors: L. Rodino and M.W. Wong, Operator Theory: Advances and Applications 189, Birkhäuser, 2008, 137–184.
C. Garetto, T. Gramchev, and M. Oberguggenberger, Pseudodifferential operators with generalized symbols and regularity theory, Electron. J. Differential Equations 2005(116) (2005), 1–43.
C. Garetto and G. Hörmann, Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities, Proc. E dinburgh Math. Soc. 48 (2005), 603–629.
C. Garetto and G. Hörmann, On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets, Bull. Cl. Sci. Math. Nat. Sci. Math. 31 (2006), 115–136.
C.Garetto, G. Hörmann, andM. Oberguggenberger, Generalized oscillatory integrals and Fourier integral operators, Proc. E dinburgh Math. Soc. 52 (2009), 351–386.
C. Garetto and M. Oberguggenberger, Fourier integral operator methods for hyperbolic equations with singularities, preprint, 2011.
C. Garetto and M. Oberguggenberger, Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients, Math. Nachrichten, to appear.
M. Grosser, M. Kunzinger, M. Oberguggenberger, and R. Steinbauer, Geometric Theory of Generalized Functions with Applications to General Relativity, Mathematics and its Applications 537, Kluwer Acad. Publ., Dordrecht, 2001.
S. Haller and G. Hörmann, Comparison of some solution concepts for linear firstorder hyperbolic differential equations with non-smooth coefficients. Publ. Inst. Math. 84(98) (2008), 123–157.
G. Hörmann and M.V. de Hoop, Microlocal analysis and global solutions of some hyperbolic equations with discontinuous coefficients, Acta Appl. Math. 67 (2001), 173–224.
G. Hörmann and M. Oberguggenberger, Elliptic regularity and solvability for partial differential equations with Colombeau coefficients, Electron. J. Differential Equations 14 (2004), 1–30.
G. Hörmann, M. Oberguggenberger and S. Pilipović, Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients, Trans. Amer. Math. Soc. 358 (2006), 3363–3383.
F. Lafon and M. Oberguggenberger, Generalized solutions to symmetric hyperbolic systems with discontinuous coefficients: the multidimensional case, J. Math. Anal. Appl. 160 (1991), 93–106.
M. Oberguggenberger, Hyperbolic systems with discontinuous coefficients: generalized solutions and a transmission problem in acoustics, J. Math. Anal. Appl. 142 (1989), 452–467.
M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes Math. 259, Longman Scientific & Technical, Harlow, 1992.
M. Oberguggenberger, Hyperbolic systems with discontinuous coefficients: generalized wavefront sets, in New Developments in Pseudo-Differential Operators, editors: L. Rodino and M.W. Wong, Birkhäuser, 2009, 117–136.
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Geguchi, H., Hörmann, G., Oberguggenberger, M. (2013). The Wave Equation with a Discontinuous Coefficient Depending on Time Only: Generalized Solutions and Propagation of Singularities. In: Molahajloo, S., Pilipović, S., Toft, J., Wong, M. (eds) Pseudo-Differential Operators, Generalized Functions and Asymptotics. Operator Theory: Advances and Applications, vol 231. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0585-8_18
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DOI: https://doi.org/10.1007/978-3-0348-0585-8_18
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