Initial value problems in elasticity

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1. H. D. Alber and R. Leis, [1988], Initial-boundary value and scattering problems in mathematical physics, in: Lecture Notes in Math. 1357, Springer, 23-60. Zbl0687.35065
2. A. L. Belopol'skiĭ and M. Sh. Birman, [1968], The existence of wave operators in the theory of scattering with a pair of spaces, Math. USSR-Izv. 2, 1117-1130. 
3. J. Bergh and J. Löfström , Interpolation Spaces. An Introduction, Springer, Berlin. Zbl0344.46071
4. D. Christodoulou, [1986], Global solutions of nonlinear hyperbolic equations for small data, Comm. Pure Appl. Math. 39, 267-287. Zbl0612.35090
5. G. F. D. Duff, [1960], The Cauchy problem for elastic waves in an anisotropic medium, Philos. Trans. Roy. Soc. London Ser. A 252, 249-273. Zbl0103.42502
6. D. M. Èĭdus, [1962], The principle of limiting absorption, Mat. Sb. 57 (99), 13-44 and AMS Transl. (2) 47 (1965), 157-191. 
7. A. Erdélyi, [1956], Asymptotic Expansions, Dover, New York. Zbl0070.29002
8. H. Freudenthal, [1939], Über ein Beugungsproblem aus der elektromagnetischen Lichttheorie, Compositio Math. 6, 221-227. Zbl0020.27104
9. L. Hörmander, [1976], The boundary problems of physical geodesy, Arch. Rational Mech. Anal. 62, 1-52. Zbl0331.35020
10. T. Ikebe, [1960], Eigenfunction expansions associated with the Schrödinger operators and their applications to scattering theory, ibid. 5, 1-34. Zbl0145.36902
11. S. Jiang, [1988], Global existence and asymptotic behaviour of smooth solutions in one-dimensional nonlinear thermoelasticity, thesis, University of Bonn. 
12. S. Jiang, [1990a], Far field behavior of solutions to the equations of nonlinear 1-d-thermoelasticity, Appl. Anal. 36, 25-35. Zbl0672.35011
13. S. Jiang, [1990b], Numerical solution for the Cauchy Problem in nonlinear 1-d-thermoelasticity, Computing 44, 147-158. Zbl0701.73001
14. S. Jiang and R. Racke, [1990], On some quasilinear hyperbolic-parabolic initial boundary value problems, Math. Methods Appl. Sci. 12, 315-339. Zbl0706.35098
15. F. John, [1974], Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27, 377-405. Zbl0302.35064
16. F. John, [1976], Delayed singularity formation in solutions of nonlinear wave equations in higher dimensions, ibid. 29, 649-681. 
17. F. John, [1977], Finite amplitude waves in a homogeneous isotropic elastic solid, ibid. 30, 421-446. Zbl0404.73023
18. F. John, [1981], Blow-up for quasi-linear wave equations in three space dimensions, ibid. 34, 29-51. Zbl0453.35060
19. F. John, [1983], Formation of singularities in elastic waves, in: Trends and Applications of Pure Mathematics to Mechanics, Proceedings Palaiseau 1983, P. G. Ciarlet and M. Roseau (eds.), Lecture Notes in Phys. 195, Springer, 194-210. 
20. F. John, [1986a], Long time effects of nonlinearity in second order hyperbolic equations, Comm. Pure Appl. Math. 39 (S), 139-148. 
21. F. John, [1986b], Partial Differential Equations, Appl. Math. Sci. 1, Springer, New York. 
22. F. John, [1987], Existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data, Comm. Pure Appl. Math. 40, 79-109. Zbl0662.35070
23. F. John, [1988], Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, ibid. 41, 615-666. Zbl0635.35066
24. F. John, [1990], Nonlinear Wave Equations, Formation of Singularities, Univ. Lecture Ser. No 2, Amer. Math. Soc. Providence, R.I. 
25. K. Jörgens, [1961], Das Anfangswertproblem im Großen für eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77, 295-308. Zbl0111.09105
26. T. Kato, [1970], Linear evolution equations of ``hyperbolic'' type, J. Fac. Sci. Univ. Tokyo 17, 241-258. Zbl0222.47011
27. T. Kato, [1973], Linear evolution equations of ``hyperbolic'' type, II, J. Math. Soc. Japan 25, 648-666. Zbl0262.34048
28. T. Kato, [1975a], Quasi-linear equations of evolution, with applications to partial differential equations, in: Lecture Notes in Math. 448, Springer, 25-70. 
29. T. Kato, [1975b], The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58, 181-205. Zbl0343.35056
30. T. Kato, [1976], Perturbation Theory for Linear Operators, Springer, Berlin. Zbl0342.47009
31. T. Kato, [1985], Abstract Differential Equations and Nonlinear Mixed Problems, Fermi Lectures. Scuola Norm. Sup. Pisa. 
32. S. Kawashima, [1983], Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, thesis, Kyoto University. 
33. S. Klainerman, [1980], Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33, 43-101. Zbl0405.35056
34. S. Klainerman, [1982], Long-time behavior of solutions to nonlinear evolution equations, Arch. Rational Mech. Anal. 78, 73-98. Zbl0502.35015
35. S. Klainerman, [1985], Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38, 321-332. Zbl0635.35059
36. S. Klainerman, [1986], The null condition and global existence to nonlinear wave equations, Lectures in Appl. Math. 23, Amer. Math. Soc., 293-326. 
37. S. Klainerman and G. Ponce, [1983], Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math. 36, 133-141. Zbl0509.35009
38. V. D. Kupradze, [1934], Über das ,,Ausstrahlungsprinzip`` von A. Sommerfeld, Dokl. Akad. Nauk SSSR 1, 55-58. 
39. P. D. Lax and R. S. Phillips, [1989], Scattering Theory, Academic Press, Boston. 
40. R. Leis, [1980], Außenraumaufgaben in der linearen Elastizitätstheorie, Math. Methods Appl. Sci. 2, 379-396. Zbl0448.35068
41. R. Leis, [1981], Über das asymptotische Verhalten thermoelastischer Wellen im ℝ³, ibid. 3, 312-317. Zbl0457.73006
42. R. Leis, [1986], Initial Boundary Value Problems in Mathematical Physics, Wiley and B. G. Teubner, Stuttgart. Zbl0599.35001
43. R. Leis, [1989], Initial-boundary value problems in elasticity, in: Pitman Res. Notes Math. 216, 73-96. Zbl0686.73006
44. O. Liess, [1989], Global existence for the nonlinear equations of crystal optics, Journ. ``Équ. aux Dér. Par.'', Exp. No. V, 11 pp., École Polyt., Palaiseau. Zbl0688.35091
45. C. S. Morawetz, [1975], Decay for solutions of the exterior problem for the wave equation, Comm. Pure Appl. Math. 28, 229-264. Zbl0304.35064
46. C. S. Morawetz and D. Ludwig, [1968], An inequality for the reduced wave operator and the justification of geometrical optics, Comm. Pure Appl. Math. 21, 187-203. Zbl0157.18701
47. C. S. Morawetz, W. Ralston and W. Strauss, [1977], Decay of solutions of the wave equation outside nontrapping obstacles, ibid. 30, 447-508 and 31, 795. Zbl0372.35008
48. J. Moser, [1961], A new technique for the construction of solutions of nonlinear differential equations, Proc. Nat. Acad. Sci. U.S.A. 47, 1824-1831. Zbl0104.30503
49. C. Müller, [1952], Zur Methode der Strahlungskapazität von H. Weyl, Math. Z. 56, 80-83. Zbl0046.10705
50. C. Müller, [1954], On the behavior of the solutions of the differential equation Δu = F(x,u) in the neighborhood of a point, Comm. Pure Appl. Math. 7, 505-515. Zbl0056.32201
51. D. B. Pearson, [1978], A generalization of the Birman trace theorem, J. Funct. Anal. 28, 182-186. Zbl0382.47006
52. H. Pecher, [1976], ${ℒ}^p$- Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen I, Math. Z. 150, 159-183. Zbl0318.35054
53. R. Racke, [1988], Initial boundary value problems in thermoelasticity, in: Partial Differential Equations and Calculus of Variations, Lecture Notes in Math. 1357, Springer, 341-358. 
54. R. Racke, [1990a], ${ℒ}^p$-${ℒ}^q$ estimates for solutions to the equations of linear thermoelasticity in exterior domains, Asymptotic Anal. 3, 105-132. Zbl0724.35019
55. R. Racke, [1990b], A unique continuation principle and weak asymptotic behaviour of solutions to semilinear wave equations in exterior domains, Appl. Math. Letters 3, 53-56. Zbl0713.35014
56. R. Racke, [1990c], Blow-up in nonlinear three-dimensional thermoelasticity, Math. Methods Appl. Sci. 12, 267-273. Zbl0705.35081
57. R. Racke, [1990d], Decay rates for solutions of damped systems and generalized Fourier transforms, J. Reine Angew. Math. 412, 1-19. Zbl0718.35015
58. R. Racke, [1990e], On the Cauchy problem in nonlinear 3-d-thermoelasticity, Math. Z. 203, 649-682. Zbl0701.73002
59. R. Racke and G. Ponce, [1990], Global existence of solutions to the initial value problem for nonlinear thermoelasticity, J. Differential Equations 87, 70-83. Zbl0725.35065
60. R. Racke and S. Zheng, [1991], Global existence of solutions to a fully nonlinear fourth-order parabolic equation in exterior domains, Nonlinear Anal., to appear. Zbl0762.35042
61. M. Reed and B. Simon, [1972-79], Methods of Modern Mathematical Physics I - IV, Academic Press, New York. 
62. F. Rellich, [1943], Über das asymptotische Verhalten der Lösungen von Δu + λu = 0 in unendlichen Gebieten, Jber. Deutsch. Math.-Verein. 53, 57-65. Zbl0028.16401
63. J. Schauder, [1935], Das Anfangswertproblem einer quasilinearen hyperbolischen Differentialgleichung zweiter Ordnung in beliebiger Anzahl von unabhängigen Veränderlichen, Fund. Math. 24, 213-246. Zbl0011.35202
64. Y. Shibata and Y. Tsutsumi, [1986], On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain, Math. Z. 191, 165-199. Zbl0592.35028
65. Y. Shibata and Y. Tsutsumi, [1987], Local existence of solution for the initial boundary value problem of fully nonlinear wave equation, Nonlinear Anal. 11, 335-365. Zbl0651.35053
66. M. Stoth, [1991], Globale klassische Lösungen der quasilinearen Elastizitätsgleichungen für kubisch elastische Medien im ℝ², SFB 256 - Preprint #157, Universität Bonn. 
67. W. A. Strauss, [1989], Nonlinear Wave Equations, CBMS Regional Conf. Ser. in Math. 73, Amer. Math. Soc., Providence, R.I. 
68. B. R. Vainberg, [1989], Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York. Zbl0743.35001
69. I. N. Vekua, [1967], New Methods for Solving Elliptic Equations, North-Holland, Amsterdam. Zbl0146.34301
70. M. Vishik and O. A. Ladyzhenskaya, [1956], Boundary value problems for partial differential equations and certain classes of operator equations, Uspekhi Mat. Nauk 11 (6,72), 41-97 and AMS Transl. (2) 10, 223-281. 
71. H. Weyl, [1952], Kapazität von Strahlungsfeldern, Math. Z. 55, 187-198. Zbl0046.10706
72. C. H. Wilcox, [1975], Scattering Theory for the d'Alembert Equation in Exterior Domains, Lecture Notes in Math. 442, Springer, Berlin. Zbl0299.35002