Asymptotics for the nonlinear dissipative wave equation
Author:
Tokio Matsuyama
Journal:
Trans. Amer. Math. Soc. 355 (2003), 865899
MSC (2000):
Primary 35L05; Secondary 35L10
Published electronically:
November 1, 2002
MathSciNet review:
1938737
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We are interested in the asymptotic behaviour of global classical solutions to the initialboundary value problem for the nonlinear dissipative wave equation in the whole space or the exterior domain outside a starshaped obstacle. We shall treat the nonlinear dissipative term like , , and prove that the energy does not in general decay. Further, we can deduce that the classical solution is asymptotically free and the local energy decays at a certain rate as the time goes to infinity.
 1.
David
Gilbarg and Neil
S. Trudinger, Elliptic partial differential equations of second
order, 2nd ed., Grundlehren der Mathematischen Wissenschaften
[Fundamental Principles of Mathematical Sciences], vol. 224,
SpringerVerlag, Berlin, 1983. MR 737190
(86c:35035)
 2.
Nakao
Hayashi, Global existence of small solutions to quadratic nonlinear
wave equations in an exterior domain, J. Funct. Anal.
131 (1995), no. 2, 302–344. MR 1345034
(96f:35113), http://dx.doi.org/10.1006/jfan.1995.1091
 3.
Mitsuru
Ikawa, Hyperbolic partial differential equations and wave
phenomena, Translations of Mathematical Monographs, vol. 189,
American Mathematical Society, Providence, RI, 2000. Translated from the
1997 Japanese original by Bohdan I. Kurpita; Iwanami Series in Modern
Mathematics. MR
1756774 (2001j:35176)
 4.
O.
A. Ladyzhenskaya, The mathematical theory of viscous incompressible
flow, Second English edition, revised and enlarged. Translated from
the Russian by Richard A. Silverman and John Chu. Mathematics and its
Applications, Vol. 2, Gordon and Breach Science Publishers, New York, 1969.
MR
0254401 (40 #7610)
 5.
JacquesLouis
Lions and W.
A. Strauss, Some nonlinear evolution equations, Bull. Soc.
Math. France 93 (1965), 43–96. MR 0199519
(33 #7663)
 6.
T. Matsuyama, Asymptotic behaviour of solutions to the initialboundary value problem with an effective dissipation around the boundary, J. Math. Anal. Appl. 271 (2002), 467492.
 7.
T. Matsuyama, Asymptotic behaviour of solutions for the nonlinear dissipative wave equations, preprint (2001).
 8.
Sigeru
Mizohata, The theory of partial differential equations,
Cambridge University Press, New York, 1973. Translated from the Japanese by
Katsumi Miyahara. MR 0599580
(58 #29033)
 9.
Kiyoshi
Mochizuki, Decay and asymptotics for wave equations with
dissipative term, (Kyoto Univ., Kyoto, 1975) Springer, Berlin,
1975, pp. 486–490. Lecture Notes in Phys., 39. MR 0600342
(58 #29089)
 10.
K. Mochizuki, Scattering theory for wave equations (in Japanese), Kinokuniya, 1984.
 11.
Kiyoshi
Mochizuki and Takahiro
Motai, On energy decaynondecay problems for wave equations with
nonlinear dissipative term in 𝑅^{𝑁}, J. Math. Soc.
Japan 47 (1995), no. 3, 405–421. MR 1331322
(96c:35122), http://dx.doi.org/10.2969/jmsj/04730405
 12.
Kiyoshi
Mochizuki and Hideo
Nakazawa, Energy decay and asymptotic behavior of solutions to the
wave equations with linear dissipation, Publ. Res. Inst. Math. Sci.
32 (1996), no. 3, 401–414. MR 1409795
(97g:35101), http://dx.doi.org/10.2977/prims/1195162849
 13.
Cathleen
S. Morawetz, Exponential decay of solutions of the wave
equation, Comm. Pure Appl. Math. 19 (1966),
439–444. MR 0204828
(34 #4664)
 14.
Takahiro
Motai and Kiyoshi
Mochizuki, On asymptotic behaviors for wave equations with a
nonlinear dissipative term in 𝐑^{𝐍}, Hokkaido Math. J.
25 (1996), no. 1, 119–135. MR 1376496
(96m:35226)
 15.
Mitsuhiro
Nakao, Existence of global classical solutions of the
initialboundary value problem for some nonlinear wave equations, J.
Math. Anal. Appl. 146 (1990), no. 1, 217–240.
MR
1041212 (91d:35141), http://dx.doi.org/10.1016/0022247X(90)90343E
 16.
Mitsuhiro
Nakao, Stabilization of local energy in an exterior domain for the
wave equation with a localized dissipation, J. Differential Equations
148 (1998), no. 2, 388–406. MR 1643195
(2000c:35141), http://dx.doi.org/10.1006/jdeq.1998.3468
 17.
Jerome
Sather, The existence of a global classical solution of the
initialboundary value problem for
𝑐𝑚𝑢+𝑢³=𝑓, Arch. Rational
Mech. Anal. 22 (1966), 292–307. MR 0197965
(33 #6124)
 18.
Jalal
Shatah, Global existence of small solutions to nonlinear evolution
equations, J. Differential Equations 46 (1982),
no. 3, 409–425. MR 681231
(84g:35036), http://dx.doi.org/10.1016/00220396(82)901024
 19.
Yoshihiro
Shibata and Yoshio
Tsutsumi, Global existence theorem for nonlinear wave equation in
exterior domain, Recent topics in nonlinear PDE (Hiroshima, 1983)
NorthHolland Math. Stud., vol. 98, NorthHolland, Amsterdam, 1984,
pp. 155–196. MR 839275
(87f:35161), http://dx.doi.org/10.1016/S03040208(08)714981
 20.
Yoshihiro
Shibata and Yoshio
Tsutsumi, On a global existence theorem of small amplitude
solutions for nonlinear wave equations in an exterior domain, Math. Z.
191 (1986), no. 2, 165–199. MR 818663
(87i:35122), http://dx.doi.org/10.1007/BF01164023
 1.
 D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., SpringerVerlag, 1983. MR 86c:35035
 2.
 N. Hayashi, Global existence of small solutions to quadratic nonlinear wave equations in an exterior domain, J. Funct. Anal. 131 (1995), 302344. MR 96f:35113
 3.
 M. Ikawa, Hyperbolic partial differential equations and wave phenomena, Transl. Math. Monogr., Vol. 189, Amer. Math. Soc., 2000. MR 2001j:35176
 4.
 O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Revised 2nd ed., New York: Gordon and Breach, 1969. MR 40:7610
 5.
 J. L. Lions and W. A. Strauss, Some nonlinear evolution equations, Bull. Soc. Math. France 93 (1965), 4396. MR 33:7663
 6.
 T. Matsuyama, Asymptotic behaviour of solutions to the initialboundary value problem with an effective dissipation around the boundary, J. Math. Anal. Appl. 271 (2002), 467492.
 7.
 T. Matsuyama, Asymptotic behaviour of solutions for the nonlinear dissipative wave equations, preprint (2001).
 8.
 S. Mizohata, The theory of partial differential equations, Cambridge Univ. Press, 1973. MR 58:29033
 9.
 K. Mochizuki, Decay and asymptotics for wave equations with dissipative term, Lecture Notes in Phys. 39, 1975, SpringerVerlag, pp. 486490. MR 58:29089
 10.
 K. Mochizuki, Scattering theory for wave equations (in Japanese), Kinokuniya, 1984.
 11.
 K. Mochizuki and T. Motai, On energy decaynondecay problems for the wave equations with nonlinear dissipative term in , J. Math. Soc. Japan 47 (1995), 405421. MR 96c:35122
 12.
 K. Mochizuki and H. Nakazawa, Energy decay and asymptotic behavior of solutions to the wave equations with linear dissipation, Publ. RIMS, Kyoto Univ. 32 (1996), 401414. MR 97g:35101
 13.
 C. Morawetz, Exponential decay of solutions of the wave equations, Comm. Pure Appl. Math. 19 (1966), 439444. MR 34:4664
 14.
 T. Motai and K. Mochizuki, On asymptotic behaviors for wave equations with a nonlinear dissipative term in , Hokkaido Math. J. 25 (1996), 119135. MR 96m:35226
 15.
 M. Nakao, Existence of global classical solutions of the initialboundary value problem for some nonlinear wave equations, J. Math. Anal. Appl. 146 (1990), 217240. MR 91d:35141
 16.
 M. Nakao, Stabilization of local energy in an exterior domain for the wave equation with a localized dissipation, J. Differential Equations 148 (1998), 388406. MR 2000c:35141
 17.
 J. Sather, The existence of a global classical solution of the initialboundary value problem for , Arch. Rational Mech. Anal. 22 (1966), 129135. MR 33:6124
 18.
 J. Shatah, Global existence of small solutions to nonlinear evolution equations, J. Differential Equations 46 (1982), 409425. MR 84g:35036
 19.
 Y. Shibata and Y. Tsutsumi, Global existence theorem of nonlinear wave equations in the exterior domain, Lecture Notes in Num. Appl. Anal. 6 (1983), 155196, Kinokuniya/NorthHolland. MR 87f:35161
 20.
 Y. Shibata and Y. Tsutsumi, On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain, Math. Z. 191 (1986), 165199. MR 87i:35122
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
35L05,
35L10
Retrieve articles in all journals
with MSC (2000):
35L05,
35L10
Additional Information
Tokio Matsuyama
Affiliation:
Department of Mathematics, Tokai University, Hiratsuka, Kanagawa 2591292, Japan
Email:
matsu@sm.utokai.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994702031471
PII:
S 00029947(02)031471
Received by editor(s):
November 6, 2001
Received by editor(s) in revised form:
July 4, 2002
Published electronically:
November 1, 2002
Additional Notes:
Supported in part by a GrantinAid for Scientific Research (C)(2)(No.11640213), Japan Society for the Promotion of Science.
The author would like to express his sincere gratitude to Professors K. Mochizuki, M. Nakao and M. Yamaguchi for several useful comments. He is also indebted to Professors Y. Shibata, N. Hayashi and T. Kobayashi, who pointed out the uniform decay estimate to him. The author thanks Doctor H. Nakazawa for advising him of the existence of scattering states. The author also thanks the referee for a careful reading of the manuscript.
Dedicated:
Dedicated to Professor Kunihiko Kajitani on the occasion of his sixtieth birthday
Article copyright:
© Copyright 2002 American Mathematical Society
