Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 084, 22 pages      arXiv:0812.2191      https://doi.org/10.3842/SIGMA.2008.084
Contribution to the Special Issue on Dunkl Operators and Related Topics

Dunkl Hyperbolic Equations

Hatem Mejjaoli
Faculty of Sciences of Tunis, Department of Mathematics, 1060 Tunis, Tunisia

Received May 10, 2008, in final form November 24, 2008; Published online December 11, 2008

Abstract
We introduce and study the Dunkl symmetric systems. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite speed of it. Next the semi-linear Dunkl-wave equations are also studied.

Key words: Dunkl operators; Dunkl symmetric systems; energy estimates; finite speed of propagation; Dunkl-wave equations with variable coefficients.

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References

  1. Ben Saïd S., Ørsted B., The wave equation for Dunkl operators, Indag. Math. (N.S.) 16 (2005), 351-391.
  2. Brenner P., On the existence of global smooth solutions of certain semi-linear hyperbolic equations, Math. Z. 167 (1979), 99-135.
  3. Brenner P., von Wahl W., Global classical solutions of nonlinear wave equations, Math. Z. 176 (1981), 87-121.
  4. Browder F.E., On nonlinear wave equations, Math. Z. 80 (1962), 249-264.
  5. Chazarain J., Piriou A., Introduction to the theory of linear partial differential equations, Studies in Mathematics and its Applications, Vol. 14, North-Holland Publishing Co., Amsterdam - New York, 1982.
  6. Courant R., Friedrichs K., Lewy H., Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100 (1928), 32-74.
  7. Dunkl C.F., Differential-difference operators associated to reflection group, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  8. Dunkl C.F., Integral kernels with reflection group invariant, Canad. J. Math. 43 (1991), 1213-1227.
  9. Dunkl C.F., Hankel transforms associated to finite reflection groups, Contemp. Math. 138 (1992), 123-138.
  10. de Jeu M.F.E., The Dunkl transform, Invent. Math. 113 (1993), 147-162.
  11. Friedrichs K., The identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc. 55 (1944), 132-151.
  12. Friedrichs K., On the differentiability of the solutions of linear elliptic differential equations, Comm. Pure Appl. Math. 6 (1953), 299-325.
  13. Friedrichs K., Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 (1954), 345-392.
  14. Friedrichs K., Symmetric positive linear hyperbolic differential equations, Comm. Pure Appl. Math. 11 (1958), 333-418.
  15. Friedrichs K., Lax P.D., Boundary value problems for first order operators, Comm. Pure Appl. Math. 18 (1965), 365-388.
  16. Friedrichs K., Lax P.D., On symmetrizable differential operators, in Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, 128-137.
  17. Hadamard J., Sur l'intégrale résiduelle, Bull. Soc. Math. France 28 (1900), 69-90.
  18. Kreiss H.-O., Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277-298.
  19. Lax P.D., On Cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl. Math. 8 (1955), 615-633.
  20. Lax P.D., Phillips R.S., Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math. 13 (1960), 427-455.
  21. Majda A., Compressible fluid flow and systems of conservation laws in several space varaibles, Applied Mathematical Sciences, Vol. 53, Springer-Verlag, New York, 1984.
  22. Mejjaoli H., Trimèche K., Hypoellipticity and hypoanaliticity of the Dunkl Laplacian operator, Integral Transforms Spec. Funct. 15 (2004), 523-548.
  23. Mejjaoli H., Littlewood-Paley decomposition associated with the Dunkl operators, and paraproduct operators, J. Inequal. Pure Appl. Math., to appear.
  24. Rauch J., L2 is a continuable initial condition for Kreiss' mixed problems, Comm. Pure Appl. Math. 25 (1972), 265-285.
  25. Rösler M., Positivity of Dunkl's intertwining operator, Duke. Math. J. 98 (1999), 445-463, q-alg/9710029.
  26. Rösler M., A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc. 355 (2003), 2413-2438, math.CA/0210137.
  27. Schauder J., Das Anfangswertproblem einer quasilinearen hyperbolischen Differentialgleichungen zweiter Ordnung in beliebiger Anzahl von unabhängigen Veränderlichen, Fund. Math. 24 (1935), 213-246.
  28. Shirota T., On the propagation speed of hyperbolic operator with mixed boundary conditions, J. Fac. Sci. Hokkaido Univ. Ser. I 22 (1972), 25-31.
  29. Strauss W., On weak solutions of semi-linear hyperbolic equations, An. Acad. Brasil. Ci. 42 (1970), 645-651.
  30. Thangavelu S., Xu Y., Convolution operator and maximal functions for Dunkl transform, J. Anal. Math. 97 (2005), 25-56, math.CA/0403049.
  31. Trimèche K., The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual, Integral Transform. Spec. Funct. 12 (2001), 349-374.
  32. Trimèche K., Paley-Wiener theorems for Dunkl transform and Dunkl translation operators, Integral Transforms Spec. Funct. 13 (2002), 17-38.
  33. Weber H., Die partiellen Differentialgleichungen der mathematischen Physik nach Riemann's Vorlesungen in Vierter auflage, Neu Bearbeitet, Braunschweig, Friederich Vieweg, 1900.
  34. Zaremba S., Sopra un theorema d'unicità relativo alla equazione delle onde sferiche, Rend. Accad. Naz. Lincei Ser. 5 24 (1915), 904-908.


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