The Strichartz estimates for the damped wave equation and the behavior of solutions for the energy critical nonlinear equation

  • Takahisa InuiEmail author


For the linear damped wave equation (DW), the \(L^p\)\(L^q\) type estimates have been well studied. Recently, Watanabe (RIMS Kôkyûroku Bessatsu B 63:77–101, 2017) showed the Strichartz estimates for DW when \(d=2,3\). In the present paper, we give Strichartz estimates for DW in higher dimensions. Moreover, by applying the estimates, we give the local well-posedness of the energy critical nonlinear damped wave equation (NLDW) \(\partial _t^2 u - \Delta u +\partial _t u = |u|^{\frac{4}{d-2}}u\), \((t,x) \in [0,T) \times {\mathbb {R}}^d\), where \(3 \le d \le 5\). Especially, we show the small data global existence for NLDW. In addition, we investigate the behavior of the solutions to NLDW. Namely, we give a decay result for solutions with finite Strichartz norm and a blow-up result for solutions with negative Nehari functional.


Damped wave equation Dissipation Strichartz estimates Energy critical 

Mathematics Subject Classification

35L71 35A01 35B40 35B44 



The author would like to express deep appreciation to Professor Masahito Ohta and Professor Yuta Wakasugi for many useful suggestions, valuable comments and warm-hearted encouragement. The author was partially supported by JSPS Grant-in-Aid for Early-Career Scientists JP18K13444.


  1. 1.
    Brenner, P.: On \(L^p\)\(L^{p^{\prime }}\) estimates for the wave-equation. Math. Z. 145(3), 251–254 (1975)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Giga, Y.: Solutions for semilinear parabolic equations in \(L^p\) and regularity of weak solutions of the Navier–Stokes system. J. Differ. Equ. 62(2), 186–212 (1986)CrossRefGoogle Scholar
  3. 3.
    Ginibre, J., Soffer, A., Velo, G.: The global Cauchy problem for the critical nonlinear wave equation. J. Funct. Anal. 110(1), 96–130 (1992)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ginibre, J., Velo, G.: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133(1), 50–68 (1995)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Loukas, G.: Classical Fourier Analysis. Graduate Texts in Mathematics, vol. 249, 3rd edn. Springer, New York (2014)Google Scholar
  6. 6.
    Gustafson, S., Roxanas, D.: Global, decaying solutions of a focusing energy-critical heat equation in \({\mathbb{R}}^4\). J. Differ. Equ. 264(9), 5894–5927 (2018)CrossRefGoogle Scholar
  7. 7.
    Hosono, T., Ogawa, T.: Large time behavior and \(L^p\)\(L^q\) estimate of solutions of 2-dimensional nonlinear damped wave equations. J. Differ. Equ. 203(1), 82–118 (2004)CrossRefGoogle Scholar
  8. 8.
    Masahiro, I., Takahisa, I.: A Remark on Non-existence Results for the Semi-linear Damped Klein–Gordon Equations, Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kôkyûroku Bessatsu. Research Institute for Mathematical Sciences (RIMS), Kyoto (2016)Google Scholar
  9. 9.
    Ikeda, M., Inui, T., Okamoto, M., Wakasugi, Y.: \(L^p\)\(L^q\) estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Commun. Pure Appl. Anal. 18(4), 1967–2008 (2019)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Masahiro, I., Yuta, W.: Global well-posedness for the semilinear wave equation with time dependent damping in the overdamping case (preprint). arXiv:1708.08044
  11. 11.
    Takahisa, I., Yuta, W.: Endpoint Strichartz estimate for the damped wave equation and its application (preprint). arXiv:1903.05891
  12. 12.
    Lev, K.: The Cauchy problem for the semilinear wave equation. I, (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 163 (1987), Kraev. Zadachi Mat. Fiz. i Smezhn. Vopr. Teor. Funktsiĭ 19, 76–104, 188; translation in J. Soviet Math. 49(5), 1166–1186 (1990)Google Scholar
  13. 13.
    Lev, K.: The Cauchy problem for the semilinear wave equation. II, (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 182 (1990), Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsiĭ. 21, 38–85, 171; translation in J. Soviet Math. 62(3), 2746–2777 (1992)Google Scholar
  14. 14.
    Lev, K.: The Cauchy problem for the semilinear wave equation. III (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 181 (1990), Differentsial’naya Geom. Gruppy Li i Mekh. 11, 24–64, 186; translation in J. Soviet Math. 62(2), 2619–2645 (1992)Google Scholar
  15. 15.
    Kapitanskiĭ, L.: Global and unique weak solutions of nonlinear wave equations. Math. Res. Lett. 1(2), 211–223 (1994)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kenig, C.E., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201(2), 147–212 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Keel, M., Tao, T.: Endpoint Strichartz estimates. Am. J. Math. 120(5), 955–980 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Herbert, K., Daniel, T., Monica, V.: Dispersive Equations and Nonlinear Waves. Generalized Korteweg–de Vries, Nonlinear Schrödinger, Wave and Schrödinger Maps, Oberwolfach Seminars, vol. 45. Birkhäuser, Basel (2014)zbMATHGoogle Scholar
  19. 19.
    Levine, H.A.: Some nonexistence and instability theorems for solutions of formally parabolic equations of the form \(P u_t = Au + F (u)\). Arch. Rational Mech. Anal. 51, 371–386 (1973)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Li, T.T., Zhou, Y.: Breakdown of solutions to \(\square u + u_t =|u|^{1+\alpha }\). Discrete Contin. Dyn. Syst. 1(4), 503–520 (1995)CrossRefGoogle Scholar
  21. 21.
    Matsumura, A.: On the asymptotic behavior of solutions of semi-linear wave equations. Publ. Res. Inst. Math. Sci. 12(1), 169–189 (1976/1977)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Takashi, N.: \(L^p\)\(L^q\) estimates for damped wave equations with odd initial data. Electron. J. Differ. Equ. 74, 17 (2005)zbMATHGoogle Scholar
  23. 23.
    Nishihara, K.: \(L^p\)\(L^q\) estimates of solutions to the damped wave equation in 3-dimensional space and their application. Math. Z. 244(3), 631–649 (2003)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ohta, M.: Blowup of solutions of dissipative nonlinear wave equations. Hokkaido Math. J. 26(1), 115–124 (1997)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Payne, L.E., Sattinger, D.H.: Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. 22(3–4), 273–303 (1975)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Pecher, H.: Nonlinear small data scattering for the wave and Klein–Gordon equation. Math. Z. 185(2), 261–270 (1984)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Shatah, J., Struwe, M.: Regularity results for nonlinear wave equations. Ann. Math. (2) 138(3), 503–518 (1993)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Souplet, P.: Nonexistence of global solutions to some differential inequalities of the second order and applications. Port. Math. 52, 289–299 (1995)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4) 110, 353–372 (1976)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Watanabe, T.: Strichartz type estimates for the damped wave equation and their application. RIMS Kôkyûroku Bessatsu B 63, 77–101 (2017)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Weissler, F.B.: Existence and nonexistence of global solutions for a semilinear heat equation. Israel J. Math. 38(1–2), 29–40 (1981)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonakaJapan

Personalised recommendations