During the last decade, much has been done in the study of singularity propagation for nonlinear partial differential equations, which has already been introduced in [1] and [6]. However, one has not seen much work on the reflection of singularities at boundary before. In 1979, M. Reed & J. Berning [8] proved that the singularities still propagate along the characteristic curves after reflection at the boundary for semilinear wave equations in one-dimensional case. For the multi-dimensional case, there have been a lot of work lately, such as that done by G. Métivier [6]，M. Beals & G. Métivier [2] and M. Sablé-Tougeron [9], who use the tools of pseudodifferential and paradifferential operators in conormal distributions. Nevertheless, one has not seen any work done by classical methods in piecewise smooth solutions up till now, while to solve this problem is of equal importance.
On the other hand, J. Rauch & M. Reed [7) proved the following result about the singularity propagation of the solution to the Cauchy problem for semilinear hyperbolic systems by classical methods:
Given a symmetric strictly hyperbolic system ...
...
Thanks to the regularity of the boundary, we can flatten it locally by a transformation of the independent variables. So we suppose the boundary to be ...
To get the a priori estimate of the solution, we must estimate the jumps of the solution across the characteristic hypersurface ...
For convenience, we define a norm for piecewise smooth functions:
...
Finally, we give the proof of the later part of the main theorem.
...
which is a contradiction.
[1] Beals M., Presence and absence of weak singularities in nonlinear waves; Preprint.
[2] Beals M. &. Métivier G., Reflection of transversal progressing waves in nonlinear
strictly hyperbolic mixed problem; To appear.
[3]
Chazarain J. & Piriou A., Introduction to the Theory of Linear Partial Differential
Equations; North-Holland Publishing Company, 1982.
[4]
Chen Shuxing. On the initial-boundary value problems for quasilinear symmetric
hyperbolic system and their applications (in Chinese); Chinese Annals of Mathematics,
1(3.4) 1980.
[5]
Chen Shuxing, Piecewise smooth solutions of semilinear hyperbolic systems in high
space dimension; To appear.
[6]
Métivier G., Problème de Cauchy et ondes non linéaires; Saint-Jean-De-Mont, 2-6
1986.
[7]
Rauch J. & Reed M., Discontinuous progressing waves for semilinear systems;
Comm. P. D. E., 10 (1985), 1033-1075.
[8]
Reed M. &. Berning J., Reflection of singularties of one-dimensional semilinear
wave equation at boundary; J. Math. Anal. Appl., 72(1979), 635-653.
[9]
Sablé-Tougeron M., Regularité microlocale pour des problèmes aux limites non
linéaires; To appear.
[10] Chen Shuxing, A Biref Introduction to the Theory of Partial Differential Equations
(in Chinese); People's Education Press, 1981.
[11] Gu Chaohao, Differentiable solutions of symmetric positive partial differential
equations; Chinese J. Math., 5 (1964), 541-545.
[12] Gu Chaohao, Some developments and applications of the theory of symmetric
positive partial differential equations; Selected papers in mathematics, Fudan
Institute of Mathematics, Fudan Univ., (1964)，42-58.
"Reflection of Singularities at Boundary for Piecewise Smooth Solutions to Semilinear Hyperbolic Systems" （“半线性双曲组的分片光滑解在边界上的奇性反射”）, Journal of Partial Differential Equations（《偏微分方程》杂志）, Vol. 2, No.1, (1989), pp. 59-70.