Local decay of solutions of conservative first order hyperbolic systems in odd dimensional space
HTML articles powered by AMS MathViewer
- by James V. Ralston
- Trans. Amer. Math. Soc. 194 (1974), 27-51
- DOI: https://doi.org/10.1090/S0002-9947-1974-0352714-9
- PDF | Request permission
Abstract:
This paper deals with symmetric hyperbolic systems, $\partial u/\partial t = Lu$, where L is equal to the homogeneous, constant coefficient operator ${L_0}$ for $|x| > R$. Under the hypothesis that L has simple null bicharacteristics and these propagate to infinity, local decay of solutions and completeness of the wave operators relating solutions of $\partial u/\partial t = Lu$ and solutions of $\partial u/\partial t = {L_0}u$ are established. Results of this type for elliptic L are due to Lax and Phillips. The proof here is based, in part, on a new estimate of the regularity of the ${L^2}$-solutions of the equation $Lu + (i\lambda + \varepsilon )u = g$ for smooth g with support in $|x| \leq R$.References
- D. M. Èĭdus, On the principle of limiting absorption, Mat. Sb. (N.S.) 57 (99) (1962), 13–44 (Russian). MR 0145187
- Charles Irwin Goldstein, Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries. I, Trans. Amer. Math. Soc. 135 (1969), 1–31. MR 234140, DOI 10.1090/S0002-9947-1969-0234140-4
- Lars Hörmander, Linear differential operators, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 121–133. MR 0513000
- Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 388463, DOI 10.1007/BF02392052
- Lars Hörmander, On the existence and the regularity of solutions of linear pseudo-differential equations, Enseign. Math. (2) 17 (1971), 99–163. MR 331124
- Peter D. Lax and Ralph S. Phillips, Scattering theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR 0217440
- Peter D. Lax and Ralph S. Phillips, Scattering theory, Rocky Mountain J. Math. 1 (1971), no. 1, 173–223. MR 412636, DOI 10.1216/RMJ-1971-1-1-173
- P. D. Lax and R. S. Phillips, Scattering theory for the acoustic equation in an even number of space dimensions, Indiana Univ. Math. J. 22 (1972/73), 101–134. MR 304882, DOI 10.1512/iumj.1972.22.22011
- Ralph Phillips and Leonard Sarason, Energy decay for the neutrino equation in the exterior of a torus, Arch. Rational Mech. Anal. 41 (1971), 1–17. MR 274980, DOI 10.1007/BF00250175
- John R. Schulenberger and Calvin H. Wilcox, Eigenfunction expansions and scattering theory for wave propagation problems of classical physics, Arch. Rational Mech. Anal. 46 (1972), 280–320. MR 369948, DOI 10.1007/BF00250513
- Homer F. Walker, On the null-spaces of first-order elliptic partial differential operators in $R{\bf ^{n}}$, Proc. Amer. Math. Soc. 30 (1971), 278–286. MR 280864, DOI 10.1090/S0002-9939-1971-0280864-7
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 194 (1974), 27-51
- MSC: Primary 35L45
- DOI: https://doi.org/10.1090/S0002-9947-1974-0352714-9
- MathSciNet review: 0352714