Spreading of singularities at the boundary in semilinear hyperbolic mixed problems. II. Crossing and self-spreading

Author:
Mark Williams

Journal:
Trans. Amer. Math. Soc. **311** (1989), 291-321

MSC:
Primary 35L70; Secondary 58G17

DOI:
https://doi.org/10.1090/S0002-9947-1989-0974778-2

MathSciNet review:
974778

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Abstract: The creation of anomalous singularities in solutions to nonlinear hyperbolic equations due to crossing or self-spreading in free space is by now rather well understood. In this paper we study how anomalous singularities are produced in mixed problems for semilinear wave equations on the half-space , due to crossing and self-spreading at boundary points. Several phenomena appear in the problems considered here which distinguish spreading at the boundary from spreading in free space: (1) Anomalous singularities of strength can arise when incoming singularity-bearing rays cross or self-spread at a point on the boundary. A consequence of this, announced in [14], is that the analogue of Beals's theorem fails for reflection in second-order mixed problems. Although regularity for propagates along null bicharacteristics in free space, for it does not in general reflect. (2) For nonlinear wave equations in free space, anomalous singular support is never produced by the interaction of fewer than three bicharacteristics, unless self-spreading occurs. However, anomalous singularities can arise when a pair of rays cross at a boundary point. (3) Suppose and on the boundary. For certain choices of initial data, anomalous singularities of strength arise at the boundary from three sources: interactions of incoming rays with incoming rays, incoming rays with reflected rays, and reflected rays with reflected rays. Singularities produced by the incoming-reflected interactions differ in sign from and are strictly weaker than the other two types, so some cancellations occur. As the incoming rays approach being gliding rays, the difference in strength decreases and hence the cancellations become increasingly significant.

**[1]**M. Beals,*Spreading of singularities for a semilinear wave equation*, Duke Math. J.**49**(1982), 275-286. MR**659941 (83m:35094)****[2]**-,*Self-spreading and strength of singularities for solutions to semilinear wave equations*, Ann. of Math.**118**(1983), 187-214. MR**707166 (85c:35057)****[3]**-,*Propagation of smoothness for nonlinear second-order strictly hyperbolic differential equations*, Proc. Sympos. Pure Math., vol. 43, Amer. Math. Soc., Providence, R.I., 1985, pp. 21-44. MR**812282 (87b:35107)****[4]**M. Beals and G. Metivier,*Progressing wave solutions to certain nonlinear mixed problems*, Duke Math. J.**53**(1986), 125-137. MR**835799 (87g:35140)****[5]**-,*Reflection of transversal progressing waves in nonlinear strictly hyperbolic mixed problems*, Amer. J. Math.**109**(1987), 335-359. MR**882427 (88e:35126)****[6]**J. Chazarain and A. Piriou,*Introduction to the theory of linear partial differential equations*, North-Holland, Amsterdam, 1982. MR**678605 (83j:35001)****[7]**F. David and M. Williams,*Singularities of solutions to semilinear boundary value problems*, Amer. J. Math.**109**(1987), 1087-1109. MR**919004 (89a:35009)****[8]**L. Hàrmander,*The analysis of linear partial differential operators*. III, Springer-Verlag, Berlin, 1985. MR**781536 (87d:35002a)****[9]**R. B. Melrose and J. Sjàstrand,*Singularities of boundary value problems*. II, Comm. Pure Appl. Math.**35**(1982), 129-168. MR**644020 (83h:35120)****[10]**G. Metivier,*Propagation, interaction, and reflection of discontinuous progressing waves for semilinear hyperbolic systems*, preprint, 1986. MR**987758 (90g:35097)****[11]**J. Rauch,*Singularities of solutions to semilinear wave equations*, J. Math. Pures Appl.**58**(1979), 299-308. MR**544255 (83c:35078)****[12]**J. Rauch and M. Reed,*Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension*, Duke Math. J.**49**(1982), 397-475. MR**659948 (83m:35098)****[13]**M. Sablé-Tougeron,*Régularité microlocale pour des problèmes aux limites non linéaires*, Ann. Inst. Fourier (Grenoble)**36**(1986), 39-82. MR**840713 (88b:35021)****[14]**M. Williams,*Spreading of singularities at the boundary in semilinear hyperbolic mixed problems*I:*Microlocal**regularity*, Duke Math. J.**56**(1988), 17-40. MR**932855 (89g:35071)**

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DOI:
https://doi.org/10.1090/S0002-9947-1989-0974778-2

Article copyright:
© Copyright 1989
American Mathematical Society