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Transactions of the American Mathematical Society

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Singularities produced in conormal wave interactions

Author: Linda M. Holt
Journal: Trans. Amer. Math. Soc. 347 (1995), 289-315
MSC: Primary 35L70; Secondary 35A20
MathSciNet review: 1264146
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Abstract: Three problems on the interactions of conormal waves are considered. Two are examples which demonstrate that nonlinear spreading of singularities can occur when the waves are conormal. In one case, two of the waves are tangential, and the other wave is transversal to the first two. The third result is a noninteraction theorem. It is shown that under certain conditions, no nonlinear spreading of the singularities will occur.

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  • [1] M. Beals, Self-spreading and strength of singularities for solutions to semilinear wave equations, Ann. of Math. 118 (1983), 187-214. MR 707166 (85c:35057)
  • [2] -, Singularities of conormal radially smooth solutions to nonlinear wave equations, Comm. Partial Differential Equations 13 (1988), 1355-1382. MR 956825 (89m:35137)
  • [3] -, Vector fields associated with the nonlinear interaction of progressing waves, Indiana Univ. Math. J. 37 (1988), 637-666. MR 962927 (89m:35138)
  • [4] -, Propagation and interaction of singularities in nonlinear hyperbolic problems, Birkhäuser, Boston, 1989. MR 1033737 (91a:58183)
  • [5] J. M. Bony, Second microlocalization and propagation of singularities for semilinear hyperbolic equations, Taniguchi Sympos., Katata, 1984, pp. 11-49. MR 925240 (89e:35099)
  • [6] -, Interaction des singularités pour les équations de Klein-Gordon non linéaires, Sém. Goulaouic-Meyer-Schwartz, Exp. no. 10, 1983-84.
  • [7] J. Chazarain and A. Piriou, Introduction to the theory of linear partial differential equations, North-Holland, Amsterdam, 1982. MR 678605 (83j:35001)
  • [8] L. Hörmander, The analysis of linear partial differential operators. I, Springer-Verlag, Berlin, 1983. MR 717035 (85g:35002a)
  • [9] L. Holt, Singularities produced in conormal wave interactions, Ph.D. thesis, Rutgers Univ., New Brunswick, N.J., 1991.
  • [10] R. Melrose and N. Ritter, Interaction of progressing waves for semilinear wave equations, Ark. Mat. 25 (1987), 91-114. MR 918380 (89b:35005)
  • [11] L. Nirenberg, Lectures on linear partial differential equations, C.B.M.S. Regional Conf. Ser. in Math., no. 17, Amer. Math. Soc., Providence, R.I., 1973. MR 0450755 (56:9048)
  • [12] J. Rauch, Singularities of solutions to semilinear wave equations, J. Math. Pures Appl. 58 (1979), 299-308. MR 544255 (83c:35078)
  • [13] J. Rauch and M. Reed, Propagation of singularities for semilinear hyperbolic equations in one space variable, Ann. of Math. 111 (1980), 531-552. MR 577136 (81h:35028)
  • [14] -, Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension, Duke Math. J. 49 (1982), 379-475. MR 659948 (83m:35098)
  • [15] -, Singularities produced by the nonlinear interaction of three progressing waves, Comm. Partial Differential Equations 7 (1982), 1117-1133. MR 673827 (83m:35097)
  • [16] A. Sá Barreto, Second microlocal ellipticity and propagation of conormality for semilinear wave equations, J. Funct. Anal. 102 (1991), 47-71. MR 1138837 (92h:35144)
  • [17] -, Evolution of semilinear waves with swallowtail singularities, preprint.

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