Skip to main content
SpringerLink
Log in

Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  • [Br] Brenner, P.: On space-time means and strong global solutions of nonlinear hyperbolic equations. Math. Z.201, 45–55 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  • [ENN] Ebihara, Y., Nakao, M., Nanbu, T.: On the existence of global classical solution of initial-boundary value problem for □u−u 3=f. Pac. J. Math.60, 63–70 (1975)

    MATH  MathSciNet  Google Scholar 

  • [Fo] Fowler, R.H.: Further studies of Emder's and similar differential equations. Q. J. Math.2, 259–288 (1931)

    Article  Google Scholar 

  • [Gl] Glassey, R.T.: Existence in the large for □u=F(u) in two space dimensions. Math. Z.178, 233–261 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  • [GNN] Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys.68, 209–243 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  • [Gr] Grillakis, M.G.: Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity. Ann. Math.132, 485–509 (1990)

    Article  MathSciNet  Google Scholar 

  • [GV] Ginibre, J., Velo, G.: The global Cauchy problem for the non linear Klein-Gordon equation. Math. Z.189, 487–505 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  • [Jo] John, F.: Nonlinear wave equations, formation of singularities. (Pitcher Lect. Math.) Providence, RI: Am. Math. Soc. 1989

    Google Scholar 

  • [Jö] Jörgens, K.: Das Anfangswertproblem im Großen für eine Klasse nichtlinearer Wellengleichungen. Math. Z.77, 295–308 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  • [Ka] Kato, T.: Abstract differential equations and nonlinear mixed problems. Pisa: Sc. Normal. Super. 1985

    Google Scholar 

  • [Kl] Klainerman, S.: Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions. Commun. Pure Appl. Math.38, 633–641 (1985)

    Google Scholar 

  • [Le1] Levine, H.A.: Instability and nonexistence of global solutions to nonlinear wave equations of the formPu tt= -A uβF (u). Trans. Am. Math. Soc.192, 1–21, (1974)

    Article  MATH  Google Scholar 

  • [Le2] Levine, H.A.: Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics: The method of unbounded Fourier coefficients. Math. Ann.214, 205–220 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  • [Li] Lions, P.L.: On the existence of positive solutions of semilinear elliptic equations. SIAM Rev.24, 441–467 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  • [Ma] Matsumura, A.: Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first-order dissipation. Publ. Res. Inst. Math. Sci.13, 349–379 (1977)

    MathSciNet  Google Scholar 

  • [Na1] Nakao, M.: Asymptotic stability of the bounded or almost periodic solution of the wave equation with nonlinear dissipative term. J. Math. Anal. Appl.58, 336–343 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  • [Na2] Nakao, M.: A difference inequality and its application to nonlinear evolution equations. J. Math. Soc. Japan30, 747–762 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  • [Na3] Nakao, M.: Existence, nonexistence and some asymptotic behaviour of global solutions of a nonlinear degenerate parabolic equation. Math. Rep., Coll. Gen. Educ., Kyushu Univ.14, 1–21 (1983)

    MATH  MathSciNet  Google Scholar 

  • [Na4] Nakao, M.: Energy decay of the wave equation with a nonlinear dissipative term. Funkc. Ekvacioj26, 237–250 (1983)

    MATH  MathSciNet  Google Scholar 

  • [Pe] Pecher, H.:L p-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen. I. Math. Z.150, 159–183 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  • [PS] Payne, L.E., Sattinger, D.H.: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math.22, 273–303 (1975)

    MathSciNet  Google Scholar 

  • [Sa] Sather, J.: The existence of a global classical solution of the initial-boundary value problem for □u+u 3=f. Arch. Ration. Mech. Anal22, 292–307 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  • [Sa] Sattinger, D.H.: On global solution of nonlinear hyperblic equations. Arch. Ration. Mech. Anal.30, 148–172 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  • [Se] Segal, I.: Non-linear semi-groups. Ann. Math.78, 339–364 (1963)

    Article  Google Scholar 

  • [St1] Strauss, W.A.: On continuity of functions with values in various Banach spaces. Pac. J. Math.19, 543–551 (1966)

    MATH  Google Scholar 

  • [St2] Strauss, W.A.: Nonlinear wave equations. (CBMS Reg. Conf. Ser. Math.) Providence, RI: Am. Math. Soc. 1989

    MATH  Google Scholar 

  • [Str] Struwe, M.: Semilinear wave equations. Bull. Am. Math. Soc.26, 53–85 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  • [Wa] Wahl, W. v.: Klassische Lösungen nichtlinearer Wellengleichungen im Großen. Math. Z.112, 241–279 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  • [Ya] Yanagida, E.: Uniqueness of positive radial solutions of Δu+g(r)u+h(r)u p=0 inR n. Arch. Ration. Mech. Anal.115, 257–274 (1991)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, College of General Education, Kyushu University, 810, Fukuoka, Japan

    Mitsuhiro Nakao

  2. Department of Mathematics, Faculty of Science, Kyushu University, 812, Fukuoka, Japan

    Kosuke Ono

Authors
  1. Mitsuhiro Nakao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kosuke Ono
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Professor J. Kajiwara on the occasion of his sixtieth birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nakao, M., Ono, K. Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations. Math Z 214, 325–342 (1993). https://doi.org/10.1007/BF02572407

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02572407

Keywords

Search

Navigation