Damped wave equation with a critical nonlinearity
Authors:
Nakao Hayashi, Elena I. Kaikina and Pavel I. Naumkin
Journal:
Trans. Amer. Math. Soc. 358 (2006), 11651185
MSC (2000):
Primary 35Q55; Secondary 35B40
Published electronically:
April 22, 2005
MathSciNet review:
2187649
Fulltext PDF Free Access
Abstract 
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Abstract: We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity
where and space dimensions . Assume that the initial data where weighted Sobolev spaces are Also we suppose that where Then we prove that there exists a positive such that the Cauchy problem above has a unique global solution satisfying the time decay property for all where
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 1.
 H. Fujita, On the blowingup of solutions of the Cauchy problem for J. Fac. Sci. Univ. of Tokyo, Sect. I, 13 (1966), 109124. MR 0214914 (35:5761)
 2.
 V.A. Galaktionov, S.P. Kurdyumov and A.A. Samarskii, On asymptotic eigenfunctions of the Cauchy problem for a nonlinear parabolic equation, Math. USSR Sbornik, 54 (1986), 421455.
 3.
 K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503505. MR 0338569 (49:3333)
 4.
 N. Hayashi, E.I. Kaikina and P.I. Naumkin, Large time behavior of solutions to the dissipative nonlinear Schrödinger equation, Proceedings of the Royal Soc. Edingburgh, 130A (2000), 10291043. MR 1800091 (2001j:35253)
 5.
 N. Hayashi, E.I. Kaikina and P.I. Naumkin, Global existence and time decay of small solutions to the LandauGinzburg type equations, Journal d'Analyse Mathematique 90 (2003), 141173. MR 2001068 (2004g:35116)
 6.
 R. Ikehata and M. Ohta, Critical exponents for semilinear dissipative wave equations in , J. Math. Anal. Appl. 269 (2002), 8797. MR 1907875 (2003i:35193)
 7.
 G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math., 143 (2) (2000), 175197. MR 1813366 (2001k:35209)
 8.
 S. Kawashima, M. Nakao, K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan, 47 (4) (1995), 617653. MR 1348752 (96g:35126)
 9.
 K. Kobayashi, T. Sirao and H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407424. MR 0450783 (56:9076)
 10.
 T.T. Li and Y. Zhou, Breakdown of solutions to , Discrete Contin. Dynam. Systems, 1 (4) (1995), 503520. MR 1357291 (96m:35222)
 11.
 A. Matsumura, On the asymptotic behavior of solutions of semilinear wave equations, Publ. Res. Inst. Math. Sci., Kyoto Univ., 12 (1976), 169189. MR 0420031 (54:8048)
 12.
 T. Narazaki, estimates for the damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan, 56 (2004), 585626. MR 2048476
 13.
 K. Nishihara, estimates of solutions to the damped wave equation in 3dimensional space and their application, Math. Z, 244 (2003), 631649. MR 1992029
 14.
 K. Nishihara, estimates for the 3D damped wave equation and their application to the semilinear problem, Seminor Notes of Math. Sci. 6 (2003), 6983, Ibaraki University.
 15.
 K. Nishitani and H. Zhao, Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption, preprint (2004).
 16.
 K. Ono, Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations, Discrete and Continuous Dynamical. Systems, 9 (2003), 651662. MR 1974531 (2004b:35237)
 17.
 G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, C.R. Acad. Sci. Paris, Série I, 330 (2000) 557562. MR 1760438 (2001a:35025)
 18.
 G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Diff. Equations, 174 (2001), 464489. MR 1846744 (2002k:35218)
 19.
 Q.S. Zhang, A blowup result for a nonlinear wave equation with damping: The critical case, C.R. Acad. Sci. Paris, Série I, 333 (2001), 109114. MR 1847355 (2003d:35189)
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Additional Information
Nakao Hayashi
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka, 5600043, Japan
Email:
nhayashi@math.wani.osakau.ac.jp
Elena I. Kaikina
Affiliation:
Departamento de Ciencias Básicas, Instituto Tecnológico de Morelia, Morelia CP 58120, Michoacán, Mexico
Email:
ekaikina@matmor.unam.mx
Pavel I. Naumkin
Affiliation:
Instituto de Matemáticas, UNAM Campus Morelia, AP 613 (Xangari), Morelia CP 58089, Michoacán, Mexico
Email:
pavelni@matmor.unam.mx
DOI:
http://dx.doi.org/10.1090/S0002994705038183
PII:
S 00029947(05)038183
Keywords:
Damped wave equation,
large time asymptotics
Received by editor(s):
April 1, 2003
Received by editor(s) in revised form:
April 22, 2004
Published electronically:
April 22, 2005
Additional Notes:
The second and the third authors were supported in part by CONACYT
Article copyright:
© Copyright 2005
American Mathematical Society
