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Interaction of progressing waves for semilinear wave equations. II. (English) Zbl 0653.35058
[For part I, see Ann. Math., II. Ser. 121, 187-213 (1985; Zbl 0575.35063).]
From the authors’ summary: “This paper is related to the propagation of conormal regularity for solutions to semilinear wave equations, i.e. to the interaction of progressing waves for such an equation. One result of this type is proved here, for the wave operator in three dimensional space-time, concerning propagation of singularities associated to two or more characteristic surfaces, simply tangent along a common line. This special case is analysed in considerable detail for several reasons but principally to check the usefulness of different notions of regularity at such a singular variety. Three distinct spaces of iterated regularity associated to this geometry are investigated.”
Reviewer: A.D.Osborne

MSC:
35L67 Shocks and singularities for hyperbolic equations
35A20 Analyticity in context of PDEs
35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
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References:
[1] Beals, M., Nonlinear wave equations with data singular at one point (preprint). · Zbl 0552.35055
[2] Beals, M. andReed, M., Propagation of singularities for hyperbolic pseudo differential operators with non-smooth coefficients,Comm. Pure Appl. Math. 35 (1982), 169–184. · Zbl 0482.35083 · doi:10.1002/cpa.3160350203
[3] Bony, J.-M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles nonlinéares,Ann. Sci. École Norm. Sup. 14 (1981), 209–246.
[4] Bony, J.-M., Interaction des singularités pour les équations aux dérivées partielles non linéares,Sem. Goulaouic-Schwartz, 1979–1980, #22 andSem. Goulaouic-Meyer-Schwartz 1981–1982, #2.
[5] Bony, J.-M., Second microlocalization and propagation of singularities for semi-linear hyperbolic equations. (preprint).
[6] Melrose, R. B., Transformation of boundary problems,Acta Math. 147 (1981), 149–236. · Zbl 0492.58023 · doi:10.1007/BF02392873
[7] Melrose, R. B. andRitter, N., Interaction of nonlinear progressing waves for semilinear wave equations,Ann. of Math. 121 (1985), 187–213. · Zbl 0575.35063 · doi:10.2307/1971196
[8] Melrose, R. B. andUhlmann, G., Lagrangian intersection and the Cauchy problem,Comm. Pure Appl. Math. (1979), 483–519. · Zbl 0396.58006
[9] Rauch, J. andReed, M., Propagation of singularities for semilinear hyperbolic equations in one space variable,Ann. of Math. 111 (1980), 531–552. · Zbl 0432.35055 · doi:10.2307/1971108
[10] Rauch, J. andReed, M., Singularities produced by the nonlinear interaction of three progressing waves; examples,Comm. Partial Differential Equations 7 (1982), 1117–1133. · Zbl 0502.35060 · doi:10.1080/03605308208820246
[11] Rauch, J. andReed, M., Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension,Duke Math. J. 49 (1982), 397–475. · Zbl 0503.35055 · doi:10.1215/S0012-7094-82-04925-0
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