Local existence and estimations for a semilinear wave equation in two dimension space

Amel Atallah Baraket

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-B, Issue: 1, page 1-21
  • ISSN: 0392-4041

Abstract

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In this paper we prove a local existence theorem for a Cauchy problem associated to a semi linear wave equation with an exponential nonlinearity in two dimension space. In this problem, the first Cauchy data is equal to zero, the second is in L 2 ⁢ R 2 , radially symmetric and compactly supported. To prove this theorem, we first show a Moser-Trudinger type inequality for the linear problem and then we use a fixed point method to achieve the proof of the result.

How to cite

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Baraket, Amel Atallah. "Local existence and estimations for a semilinear wave equation in two dimension space." Bollettino dell'Unione Matematica Italiana 7-B.1 (2004): 1-21. <http://eudml.org/doc/194701>.

@article{Baraket2004,
abstract = {In this paper we prove a local existence theorem for a Cauchy problem associated to a semi linear wave equation with an exponential nonlinearity in two dimension space. In this problem, the first Cauchy data is equal to zero, the second is in $L^\{2\}(\mathbb\{R\}^\{2\})$, radially symmetric and compactly supported. To prove this theorem, we first show a Moser-Trudinger type inequality for the linear problem and then we use a fixed point method to achieve the proof of the result.},
author = {Baraket, Amel Atallah},
journal = {Bollettino dell'Unione Matematica Italiana},
keywords = {Cauchy problem},
language = {eng},
month = {2},
number = {1},
pages = {1-21},
publisher = {Unione Matematica Italiana},
title = {Local existence and estimations for a semilinear wave equation in two dimension space},
url = {http://eudml.org/doc/194701},
volume = {7-B},
year = {2004},
}

TY - JOUR
AU - Baraket, Amel Atallah
TI - Local existence and estimations for a semilinear wave equation in two dimension space
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/2//
PB - Unione Matematica Italiana
VL - 7-B
IS - 1
SP - 1
EP - 21
AB - In this paper we prove a local existence theorem for a Cauchy problem associated to a semi linear wave equation with an exponential nonlinearity in two dimension space. In this problem, the first Cauchy data is equal to zero, the second is in $L^{2}(\mathbb{R}^{2})$, radially symmetric and compactly supported. To prove this theorem, we first show a Moser-Trudinger type inequality for the linear problem and then we use a fixed point method to achieve the proof of the result.
LA - eng
KW - Cauchy problem
UR - http://eudml.org/doc/194701
ER -

References

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  5. MOSER, J., A sharp form of an inequality of N. Trudinger, Ind. Univ. Math. J., 20 (1971), 1077-1092. Zbl0213.13001MR301504
  6. NAKUMURA, M.- OZAWA, T., Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth, Math. Z., 231, No. 3 (1999), 479-487. Zbl0931.35107MR1704989
  7. SHATAH, J.- STRUWE, M., Regularity results for nonlinear wave equations, Ann. Math., 138 (1993), 503-518. Zbl0836.35096MR1247991
  8. STRAUSS, W., On weak solutions of semi-linear hyperbolic equations, Anais Acad. Brasil Cienc., 42 (1970), 645-651. Zbl0217.13104MR306715
  9. STRUWE, M., Semilinear wave equations, Bull. Amer. Math. Soc., 26 (1992), 53-86. Zbl0767.35045MR1093058
  10. TAYLOR, M. E., Partial differential equations I. Basic Theory, Applied. Math. Sciences115. Springer. Zbl0869.35002MR1395148
  11. TRUDINGER, N. S., On imbeddings into Orlicz spaces and some applications, J. Math. Mechanics, 17, 5 (1967), 473-484. Zbl0163.36402MR216286

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