Advertisement

A Toy Model of 4D Semilinear Weakly Hyperbolic Wave Equations

  • Sandra LucenteEmail author
  • Emanuele Marrone
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

In this chapter, we prove the large data almost global existence of the 4-dimensional weakly hyperbolic equation:
$$\displaystyle u_{tt}-(t_0-t)^2\varDelta u=-(t_0-t)^4|u|u\,. $$

Notes

Acknowledgements

The Authors thank the anonymous referee for the available remarks. The first Author is grateful to the organizers of Special Interest Group IGPDE in the 11th ISAAC Congress at Linneuniversitetet in Sweden.

References

  1. 1.
    P. Brenner, W. von Wahl, Global classical solution of nonlinear wave equations. Math. Z. 176, 87–121 (1981)MathSciNetCrossRefGoogle Scholar
  2. 2.
    P. D’Ancona, A note on a theorem of Jörgens. Math. Z. 218, 239–252 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    P. D’Ancona, A. Di Giuseppe, Global existence with large data for a nonlinear weakly hyperbolic equation. Math. Nachr. 231, 5–23 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    L. Fanelli, S. Lucente, The critical case for a semilinear weakly hyperbolic equation. Electron. J. Differ. Equ. 2004(101), 1–13 (2004)MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Galstian, Global existence for the one-dimensional second order semilinear hyperbolic equations. J. Math. Anal. Appl. 344, 76–98 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M.G. Grillakis, Regularity and asymptotic behavior of the wave equation with a critical nonlinearity. Ann. Math. 132, 485–509 (1990)MathSciNetCrossRefGoogle Scholar
  7. 7.
    K. Jörgens, Das Anfangswertproblem in Groem fr eine Klasse nichtlinearen Wellengleichungen. Math. Z. 77, 295–308 (1961)MathSciNetCrossRefGoogle Scholar
  8. 8.
    J. Kim, C.H. Lee, The globally regular solutions of semilinear wave equation with a critical nonlinearity. J. Korean Math. Soc. 31, 255–278 (1994)MathSciNetzbMATHGoogle Scholar
  9. 9.
    S. Lucente, On a class of semilinear weakly hyperbolic equations. Ann. Univ. Ferrara 52, 317–335 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    S. Lucente, Large data solutions for critical semilinear weakly hyperbolic equations, in Proceeding of the Conference Complex Analysis & Dynamical System VI, vol. 653 (American Mathematical Society, Providence, 2015), pp. 251–276zbMATHGoogle Scholar
  11. 11.
    M. Reissig, On L p − L q estimates for solutions of a special weakly hyperbolic equation, in Nonlinear Evolution Equations and Infinite-Dimensional Dynamical Systems, ed. by L. Ta-Tsien (World Scientific, Singapore, 1997), pp. 153–164Google Scholar
  12. 12.
    J. Shatah, M. Struwe, Regularity result for nonlinear wave equation. Ann. Math. 138, 503–518 (1993)MathSciNetCrossRefGoogle Scholar
  13. 13.
    K. Yagdjian, A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain. J. Differ. Equ. 206, 227–252 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BariBariItaly
  2. 2.Dipartimento di MatematicaUniversità degli studi di BariBariItaly

Personalised recommendations