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Second microlocal ellipticity and propagation of conormality for semilinear wave equations. (English) Zbl 0746.35021
The author considers three problems on the propagation of conormal singularities for semilinear wave equations: the evolution of two simply tangent waves, the interaction of three conormal waves, and the evolution of one wave with a cusp singularity. He uses quasihomogeneous blow-ups to reduce the geometries to normal crossing and show that the lift of the operator by these blow-down maps is elliptic in some directions of the compressed cotangent bundle of the blown-up manifold. This leads to a strengthening of the previously known results and a simplification of their proofs.

MSC:
35L65 Hyperbolic conservation laws
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
58J47 Propagation of singularities; initial value problems on manifolds
35L05 Wave equation
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[1] Arnol’d, V.I, Wave fronts evolution and equivariant Morse lemma, Comm. pure appl. math., 28, 557-582, (1976) · Zbl 0343.58003
[2] \scM. Beals, Singularities of conormal radially smooth solutions to nonlinear wave equations, preprint. · Zbl 0673.35078
[3] \scM. Beals, Regularity of nonlinear waves associated with a cusp, The IMA Volumes in Mathematics and Its Applications, Vol. 30, pp. 9-27. · Zbl 0794.35004
[4] \scJ.-M. Bony, Interaction des singularitées pour les équations aux derivées partielles nonlinéaires, in “Sem. Goulaouic-Meyer-Schwartz,” Exp. No. 22, pp. 1979-1980.
[5] Bony, J.-M, Second microlocalization and propagation of singularities for semilinear hyperbolic equations, (), 11-49
[6] Chemin, J.-Y, Interaction des trois ondes dans LES équations semilinéaires strictment hyperboliques d’ordre 2, Comm. partial differential equations, 12, No. 11, 1203-1225, (1987) · Zbl 0651.35049
[7] Hörmander, L, ()
[8] \scR. Melrose, Marked lagrangian distributions, in preparation.
[9] Melrose, R, Semilinear equations with cusp singularities, Jour. equations deriv. partielles, st. Jean de monts, (1987) · Zbl 0656.35098
[10] \scR. Melrose, Conormality, cusps and non-linear equation, The IMA Volumes in Mathematics and Its Applications, Vol. 30, pp. 155-166. · Zbl 0794.35005
[11] Melrose, R; Ritter, N, Interactions of nonlinear progressing waves for semilinear wave equations, Ann. of math., 121, 187-213, (1985) · Zbl 0575.35063
[12] Melrose, R; Ritter, N, Interaction of nonlinear progressing waves for semilinear wave equations, II, Ark. mat., 25, 91-114, (1987) · Zbl 0653.35058
[13] \scR. Melrose and A. Sá Barreto, Non-linear interaction of a cusp and a plane, in preparation. · Zbl 0847.35086
[14] \scR. Melrose, A. Sá Barreto, and M. Zworski, Semilinear diffraction of conormal waves, in preparation. · Zbl 0902.35004
[15] Ritter, N, Progressing wave solutions to nonlinear Cauchy problems, MIT thesis, (June 1984)
[16] SáBarreto, A, Interactions of conormal waves for fully semilinear wave equations, J. funct. anal., 89, 233-273, (1990)
[17] \scA. Sá Barreto, On the interactions of conormal waves, The IMA Volumes in Mathematics and Its Applications, Vol. 30, pp. 1-7. · Zbl 0794.35006
[18] \scA. Sá Barreto, Evolution of semilinear waves with swallowtail singularities, preprint. · Zbl 0827.35078
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