Abstract:We consider the creation and propagation of singularities in the solutions of semilinear nonstrictly hyperbolic systems in one space dimension when the initial data has jump discontinuities. We show that singularities travelling along characteristics can branch at points of degeneracy of the vector fields on all other forward characteristics. We prove a lower bound for the strength of these new singularities, and we give an example showing that our result cannot be improved in general.
- 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, DOI 10.2307/1971108
- Jeffrey Rauch and Michael Reed, Jump discontinuities of semilinear, strictly hyperbolic systems in two variables: creation and propagation, Comm. Math. Phys. 81 (1981), no. 2, 203–227. MR 632757
- 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
- Jeffrey Rauch and Michael C. Reed, Propagation of singularities in nonstrictly hyperbolic semilinear systems: examples, Comm. Pure Appl. Math. 35 (1982), no. 4, 555–565. MR 657827, DOI 10.1002/cpa.3160350405
- J. V. Ralston, On the propagation of singularities in solutions of symmetric hyperbolic partial differential equations, Comm. Partial Differential Equations 1 (1976), no. 2, 87–133. MR 442494, DOI 10.1080/03605307608820006
- Kazuo Taniguchi and Yoshiharu Tozaki, A hyperbolic equation with double characteristics which has a solution with branching singularities, Math. Japon. 25 (1980), no. 3, 279–300. MR 586523 L. Micheli, Ph. D. Thesis, Duke Univ., 1984.
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 291 (1985), 451-485
- MSC: Primary 35L60; Secondary 35A20, 58G17
- DOI: https://doi.org/10.1090/S0002-9947-1985-0800248-4
- MathSciNet review: 800248