On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term

Shuichi KAWASHIMA; Mitsuhiro NAKAO; Kosuke ONO

doi:10.2969/jmsj/04740617

Shuichi KAWASHIMA, Mitsuhiro NAKAO, Kosuke ONO

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JOURNAL FREE ACCESS

1995 Volume 47 Issue 4 Pages 617-653

DOI https://doi.org/10.2969/jmsj/04740617

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Published: 1995 Received: November 29, 1993 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : Br75) P. Brenner, On L_p-L_p'estimates for the wave equation, Math. Z., 145 (1975), 251-254.
Br89) P. Brenner, On space-time means and strong global solutions of nonlinear hyperbolic equations, Math. Z., 201 (1989), 45-55.
BW81) P. Brenner and W. v. Wahl, Global classical solutions of nonlinear wave equations, Math. Z., 176 (1981), 87-121.
GV85) J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505.
Gr90) M. G. Grillakis, Regularity and asymptotic behavior of the wave equation with a critical nonlinearity, Ann. of Math., 132 (1990), 485-509.
Hö60) L. Hörmander, Estimates for translation invariant operators in L_p-spaces, Acta Math., 104 (1960), 93-145.
Jö61) K. Jörgens, Das Anfangswertproblem in Großen für eine Klasse nichtlinearer Wellengleichungen, Math. Z., 77 (1961), 295-308.
Ma76) A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., Kyoto Univ., 12 (1976), 169-189.
Mi65) S. G. Michlin, Multidimensional Singular Integrals and Integral Equations, Ox-ford-London-New York-Paris, 1965.
Mo84) K. Mochizuki, Scattering Theory of The Wave Equation, Kinokuniya Shoten, 1984, (in Japanese).
Na78) M. Nakao, A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan., 30 (1978), 747-762.
Na83) M. Nakao, Energy decay of the wave equation with a nonlinear dissipative term, Funkcial. Ekvac., 26 (1983), 237-250.
Pe76) H. Pecher, L^p-Abschätzunger und Klassische Lösungen für nichtlineare Wellen-gleich ungen I, Math. Z., 150 (1976), 159-183.
Ra90) R. Racke, Decay rates for solutions of damped systems and generalized Fourier transforms, J. Reine Angew. Math., 412 (1990), 1-19.
St70) W. A. Strauss, On weak solutions of semi-linear hyperbolic equations, An. Acad. Brasil. Ciênc., 42 (1970), 645-651.

Right : [Br75] P. Brenner, On L_p-L_p' estimates for the wave equation, Math. Z., 145 (1975), 251-254.
[Br89] P. Brenner, On space-time means and strong global solutions of nonlinear hyperbolic equations, Math. Z., 201 (1989), 45-55.
[BW81] P. Brenner and W. v. Wahl, Global classical solutions of nonlinear wave equations, Math. Z., 176 (1981), 87-121.
[GV85] J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505.
[Gr90] M. G. Grillakis, Regularity and asymptotic behavior of the wave equation with a critical nonlinearity, Ann. of Math., 132 (1990), 485-509.
[Hö60] L. Hörmander, Estimates for translation invariant operators in L_p-spaces, Acta Math., 104 (1960), 93-145.
[Jö61] K. Jörgens, Das Anfangswertproblem in Großen für eine Klasse nichtlinearer Wellengleichungen, Math. Z., 77 (1961), 295-308.
[Ma76] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., Kyoto Univ., 12 (1976), 169-189.
[Mi65] S. G. Michlin, Multidimensional Singular Integrals and Integral Equations, Oxford-London-New York-Paris, 1965.
[Mo84] K. Mochizuki, Scattering Theory of The Wave Equation, Kinokuniya Shoten, 1984, (in Japanese).
[Na78] M. Nakao, A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan., 30 (1978), 747-762.
[Na83] M. Nakao, Energy decay of the wave equation with a nonlinear dissipative term, Funkcial. Ekvac., 26 (1983), 237-250.
[Pe76] H. Pecher, L^p-Abschätzunger und Klassische Lösungen für nichtlineare Wellen-gleich ungen I, Math. Z., 150 (1976), 159-183.
[Ra90] R. Racke, Decay rates for solutions of damped systems and generalized Fourier transforms, J. Reine Angew. Math., 412 (1990), 1-19.
[St70] W. A. Strauss, On weak solutions of semi-linear hyperbolic equations, An. Acad. Brasil. Ciênc., 42 (1970), 645-651.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

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