On a linear hyperbolic equation with smooth coefficients without solutions

Yuri Egorov

Banach Center Publications (1996)

  • Volume: 33, Issue: 1, page 61-66
  • ISSN: 0137-6934

Abstract

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An example of a locally unsolvable hyperbolic equation of the second order is constructed, which has smooth ( C ∞ ) coefficients, but has no solutions in the class of distributions.

How to cite

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Egorov, Yuri. "On a linear hyperbolic equation with smooth coefficients without solutions." Banach Center Publications 33.1 (1996): 61-66. <http://eudml.org/doc/262569>.

@article{Egorov1996,
abstract = {An example of a locally unsolvable hyperbolic equation of the second order is constructed, which has smooth ($C^∞$) coefficients, but has no solutions in the class of distributions.},
author = {Egorov, Yuri},
journal = {Banach Center Publications},
keywords = {locally unsolvable hyperbolic equation},
language = {eng},
number = {1},
pages = {61-66},
title = {On a linear hyperbolic equation with smooth coefficients without solutions},
url = {http://eudml.org/doc/262569},
volume = {33},
year = {1996},
}

TY - JOUR
AU - Egorov, Yuri
TI - On a linear hyperbolic equation with smooth coefficients without solutions
JO - Banach Center Publications
PY - 1996
VL - 33
IS - 1
SP - 61
EP - 66
AB - An example of a locally unsolvable hyperbolic equation of the second order is constructed, which has smooth ($C^∞$) coefficients, but has no solutions in the class of distributions.
LA - eng
KW - locally unsolvable hyperbolic equation
UR - http://eudml.org/doc/262569
ER -

References

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  1. [1] M. H. Protter, The Cauchy problem for a hyperbolic second order equation with data on the parabolic line, Canad. J. Math. 6 (1954), 542-553. Zbl0057.08101
  2. [2] M. M. Smirnov, Degenerating Elliptic and Hyperbolic Equations, Nauka, Moscow, 1966 (in Russian). 
  3. [3] V. Ya. Ivriĭ and V. Petkov, Necessary conditions for the Cauchy problem for non-strictly hyperbolic equations to be well-posed, Russian Math. Surveys 29 (1974), 1-70. 
  4. [4] O. A. Oleĭnik, On the Cauchy problem for weakly hyperbolic equations, Comm. Pure Appl. Math. 23 (1970), 569-586. 
  5. [5] H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math. 66 (1957), 155-158. Zbl0078.08104
  6. [6] L. Hörmander, Linear Partial Differential Operators, Springer, 1976. 
  7. [7] L. Nirenberg and F. Trèves, On local solvability of linear partial differential operators, I. Necessary conditions, Comm. Pure Appl. Math. 23 (1970), 1-38. Zbl0191.39103
  8. [8] Yu. V. Egorov, On the local solvability of the differential equations with simple characteristics, Russian Math. Surveys 26 (1971), 183-198. 
  9. [9] Yu. V. Egorov, Linear Differential Equations of Principal Type, Plenum, 1986. 
  10. [10] L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. III, Springer, 1985. Zbl0601.35001
  11. [11] Y. Kannai, An unsolvable hypoelliptic diferential operator, Israel Math. J. 9 (1971), 306-315. Zbl0211.40601
  12. [12] F. Colombini and S. Spagnolo, Some examples of hyperbolic equations without local solvability, Ann. Sci. Ecole Norm. Sup. 22 (1989), 109-125. Zbl0702.35146

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