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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Michael Beals,*Self-spreading and strength of singularities for solutions to semilinear wave equations*, Ann. of Math. (2)**118**(1983), no. 1, 187–214. MR**707166**, 10.2307/2006959**[2]**Michael Beals,*Singularities of conormal radially smooth solutions to nonlinear wave equations*, Comm. Partial Differential Equations**13**(1988), no. 11, 1355–1382. MR**956825**, 10.1080/03605308808820579**[3]**Michael Beals,*Vector fields associated with the nonlinear interaction of progressing waves*, Indiana Univ. Math. J.**37**(1988), no. 3, 637–666. MR**962927**, 10.1512/iumj.1988.37.37031**[4]**Michael Beals,*Propagation and interaction of singularities in nonlinear hyperbolic problems*, Progress in Nonlinear Differential Equations and their Applications, 3, Birkhäuser Boston, Inc., Boston, MA, 1989. MR**1033737****[5]**Jean-Michel Bony,*Second microlocalization and propagation of singularities for semilinear hyperbolic equations*, Hyperbolic equations and related topics (Katata/Kyoto, 1984) Academic Press, Boston, MA, 1986, pp. 11–49. MR**925240****[6]**-,*Interaction des singularités pour les équations de Klein-Gordon non linéaires*, Sém. Goulaouic-Meyer-Schwartz, Exp. no. 10, 1983-84.**[7]**Jacques Chazarain and Alain Piriou,*Introduction to the theory of linear partial differential equations*, Studies in Mathematics and its Applications, vol. 14, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French. MR**678605****[8]**Lars Hörmander,*The analysis of linear partial differential operators. I*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR**717035****[9]**L. Holt,*Singularities produced in conormal wave interactions*, Ph.D. thesis, Rutgers Univ., New Brunswick, N.J., 1991.**[10]**Richard B. Melrose and Niles Ritter,*Interaction of progressing waves for semilinear wave equations. II*, Ark. Mat.**25**(1987), no. 1, 91–114. MR**918380**, 10.1007/BF02384437**[11]**Louis Nirenberg,*Lectures on linear partial differential equations*, American Mathematical Society, Providence, R.I., 1973. Expository Lectures from the CBMS Regional Conference held at the Texas Technological University, Lubbock, Tex., May 22–26, 1972; Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 17. MR**0450755****[12]**Jeffrey Rauch,*Singularities of solutions to semilinear wave equations*, J. Math. Pures Appl. (9)**58**(1979), no. 3, 299–308. MR**544255****[13]**Jeffrey Rauch and Michael C. Reed,*Propagation of singularities for semilinear hyperbolic equations in one space variable*, Ann. of Math. (2)**111**(1980), no. 3, 531–552. MR**577136**, 10.2307/1971108**[14]**Jeffrey Rauch and Michael Reed,*Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension*, Duke Math. J.**49**(1982), no. 2, 397–475. MR**659948****[15]**Jeffrey Rauch and Michael C. Reed,*Singularities produced by the nonlinear interaction of three progressing waves; examples*, Comm. Partial Differential Equations**7**(1982), no. 9, 1117–1133. MR**673827**, 10.1080/03605308208820246**[16]**Antônio Sá Barreto,*Second microlocal ellipticity and propagation of conormality for semilinear wave equations*, J. Funct. Anal.**102**(1991), no. 1, 47–71. MR**1138837**, 10.1016/0022-1236(91)90135-R**[17]**-,*Evolution of semilinear waves with swallowtail singularities*, preprint.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35L70,
35A20

Retrieve articles in all journals with MSC: 35L70, 35A20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1264146-3

Article copyright:
© Copyright 1995
American Mathematical Society