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Critical exponent for the semilinear wave equations with a damping increasing in the far field

  • Kenji Nishihara
  • Motohiro Sobajima
  • Yuta Wakasugi
Article
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Abstract

We consider the Cauchy problem of the semilinear wave equation with a damping term
$$\begin{aligned} \left\{ \begin{array}{ll} u_{tt} - \Delta u + c(t,x) u_t = |u|^p,&{}(t,x)\in (0,\infty )\times {\mathbb {R}}^N,\\ u(0,x) = \varepsilon u_0(x), \quad u_t(0,x) = \varepsilon u_1(x),&{} x\in {\mathbb {R}}^N, \end{array}\right. \end{aligned}$$
where \(p>1\) and the coefficient of the damping term has the form
$$\begin{aligned} c(t,x) = a_0 (1+|x|^2)^{-\alpha /2} (1+t)^{-\beta } \end{aligned}$$
with some \(a_0 > 0\), \(\alpha < 0\), \(\beta \in (-1, 1]\). In particular, we mainly consider the cases
$$\begin{aligned} \alpha< 0, \beta =0 \quad \text{ or } \quad \alpha < 0, \beta = 1, \end{aligned}$$
which imply \(\alpha + \beta < 1\), namely, the damping is spatially increasing and effective. Our aim is to prove that the critical exponent is given by
$$\begin{aligned} p = 1+ \frac{2}{N-\alpha }. \end{aligned}$$
This shows that the critical exponent is the same as that of the corresponding parabolic equation
$$\begin{aligned} c(t,x) v_t - \Delta v = |v|^p. \end{aligned}$$
The global existence part is proved by a weighted energy estimates with an exponential-type weight function and a special case of the Caffarelli–Kohn–Nirenberg inequality. The blow-up part is proved by a test-function method introduced by Ikeda and Sobajima [15]. We also give an upper estimate of the lifespan.

Keywords

Semilinear damped wave equation Time and space dependent damping Critical exponent Lifespan 

Mathematics Subject Classification

Primary 35L15 Secondary 35A01 35B44 

Notes

Acknowledgements

This work was supported by JSPS KAKENHI Grant Number JP18K134450 and JP16K17625.

References

  1. 1.
    Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weights. Compos. Math. 53, 259–275 (1984)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chill, R., Haraux, A.: An optimal estimate for the difference of solutions of two abstract evolution equations. J. Differ. Equ. 193, 385–395 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    D’Abbicco, M.: The threshold of effective damping for semilinear wave equations. Math. Methods Appl. Sci. 38, 1032–1045 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    D’Abbicco, M., Lucente, S.: A modified test function method for damped wave equations. Adv. Nonlinear Stud. 13, 867–892 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    D’Abbicco, M., Lucente, S.: NLWE with a special scale invariant damping in odd space dimension. In: Discrete and Continuous Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl., pp. 312–319 (2015)Google Scholar
  6. 6.
    D’Abbicco, M., Lucente, S., Reissig, M.: Semi-linear wave equations with effective damping. Chin. Ann. Math. Ser. B 34, 345–380 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D’Abbicco, M., Lucente, S., Reissig, M.: A shift in the Strauss exponent for semilinear wave equations with a not effective damping. J. Differ. Equ. 259, 5040–5073 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fujita, H.: On the blowing up of solutions of the Cauchy problem for \(u_t=\Delta u+u^{1+\alpha }\). J. Fac. Sci. Univ. Tokyo Sec. I(13), 109–124 (1966)Google Scholar
  9. 9.
    Fujiwara, K., Ikeda, M., Wakasugi, Y.: Estimates of lifespan and blow-up rates for the wave equation with a time-dependent damping and a power-type nonlinearity, to appear in Funkcial. Ekvac. arXiv:1609.01035v2
  10. 10.
    Giga, M.-H., Giga, Y., Saal, J.: Nonlinear Partial Differential Equations, Progress in Nonlinear Differential Equations and their Applications, vol. 79. Birkhäuser, Boston (2010)zbMATHGoogle Scholar
  11. 11.
    Hayashi, N., Kaikina, E.I., Naumkin, P.I.: Damped wave equation with super critical nonlinearities. Differ. Integral Equ. 17, 637–652 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    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, 82–118 (2004)CrossRefGoogle Scholar
  13. 13.
    Hsiao, L., Liu, T.-P.: Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Commun. Math. Phys. 43, 599–605 (1992)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ikeda, M., Inui, T.: The sharp estimate of the lifespan for the semilinear wave equation with time-dependent damping, arXiv:1707.03950v1
  15. 15.
    Ikeda, M., Sobajima, M.: Upper bound for lifespan of solutions to certain semilinear parabolic, dispersive and hyperbolic equations via a unified test function method, arXiv:1710.06780v1
  16. 16.
    Ikeda, M., Sobajima, M.: Life-span of blowup solutions to semilinear wave equation with space-dependent critical damping, to appear in Funkcialaj Ekvacioj, arXiv:1709.04401v1
  17. 17.
    Ikeda, M., Sobajima, M.: Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data, to appear in Mathematische Annalen, arXiv:1709.04406v1
  18. 18.
    Ikeda, M., Sobajima, M., Wakasugi, Y.: Sharp lifespan estimates of blowup solutions to semilinear wave equations with time-dependent effective damping, arXiv:1808:06189v1
  19. 19.
    Ikeda, M., Wakasugi, Y.: A note on the lifespan of solutions to the semilinear damped wave equation. Proc. Am. Math. Soc. 143, 163–171 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ikeda, M., Wakasugi, Y.: Global well-posedness for the semilinear wave equation with time dependent damping in the overdamping case, to appear in Proc. Am. Math. SocGoogle Scholar
  21. 21.
    Ikehata, R.: Some remarks on the wave equation with potential type damping coefficients. Int. J. Pure Appl. Math. 21, 19–24 (2005)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ikehata, R., Takeda, H.: Uniform energy decay for wave equations with unbounded damping coefficients, arXiv:1706.03942v1
  23. 23.
    Ikehata, R., Tanizawa, K.: Global existence of solutions for semilinear damped wave equations in \({\mathbb{R}}^N\) with noncompactly supported initial data. Nonlinear Anal. 61, 1189–1208 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ikehata, R., Todorova, G., Yordanov, B.: Critical exponent for semilinear wave equations with space-dependent potential. Funkcialaj Ekvacioj 52, 411–435 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ikehata, R., Todorova, G., Yordanov, B.: Optimal decay rate of the energy for wave equations with critical potential. J. Math. Soc. Jpn. 65, 183–236 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Karch, G.: Selfsimilar profiles in large time asymptotics of solutions to damped wave equations. Studia Math. 143, 175–197 (2000)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kenigson, J.S., Kenigson, J.J.: Energy decay estimates for the dissipative wave equation with space–time dependent potential. Math. Methods Appl. Sci. 34, 48–62 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Khader, M.: Nonlinear dissipative wave equations with space–time dependent potential. Nonlinear Anal. 74, 3945–3963 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Khader, M.: Global existence for the dissipative wave equations with space–time dependent potential. Nonlinear Anal. 81, 87–100 (2013)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kirane, M., Qafsaoui, M.: Fujita’s exponent for a semilinear wave equation with linear damping. Adv. Nonlinear Stud. 2, 41–49 (2002)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Lai, N.A., Takamura, H.: Blow-up for semilinear damped wave equations with sub-Strauss exponent in the scattering case. Nonlinear Anal. 168, 222–237 (2018)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Lai, N.A., Takamura, H., Wakasa, K.: Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent. J. Differ. Equ. 263, 5377–5394 (2017)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Lai, N.A., Zhou, Y.: The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in higher dimensions, to appear in J. Math. Pure ApplGoogle Scholar
  34. 34.
    Li, T.-T., Zhou, Y.: Breakdown of solutions to \(\square u+u_t=|u|^{1+\alpha }\). Discrete Contin. Dynam. Syst. 1, 503–520 (1995)CrossRefGoogle Scholar
  35. 35.
    Lin, J., Nishihara, K., Zhai, J.: \(L^2\)-estimates of solutions for damped wave equations with space–time dependent damping term. J. Differ. Equ. 248, 403–422 (2010)CrossRefGoogle Scholar
  36. 36.
    Lin, J., Nishihara, K., Zhai, J.: Decay property of solutions for damped wave equations with space–time dependent damping term. J. Math. Anal. Appl. 374, 602–614 (2011)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Lin, J., Nishihara, K., Zhai, J.: Critical exponent for the semilinear wave equation with time-dependent damping. Discrete Contin. Dyn. Syst. 32, 4307–4320 (2012)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Liu, M., Wang, C.: Global existence for semilinear damped wave equations in relation with the Strauss conjecture, arXiv:1807.05908v1
  39. 39.
    Marcati, P., Nishihara, K.: The \(L^p\)-\(L^q\) estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media. J. Differ. Equ. 191, 445–469 (2003)CrossRefGoogle Scholar
  40. 40.
    Matsumura, A.: On the asymptotic behavior of solutions of semi-linear wave equations. Publ. Res. Inst. Math. Sci. 12, 169–189 (1976)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Michihisa, H.: \(L^2\)-asymptotic profiles of solutions to linear damped wave equations, arXiv:1710.04870v1
  42. 42.
    Mochizuki, K.: Scattering theory for wave equations with dissipative terms. Publ. Res. Inst. Math. Sci. 12, 383–390 (1976)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Mochizuki, K., Nakazawa, H.: Energy decay and asymptotic behavior of solutions to the wave equations with linear dissipation. Publ. RIMS Kyoto Univ. 32, 401–414 (1996)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Narazaki, T.: \(L^p\)-\(L^q\) estimates for damped wave equations and their applications to semi-linear problem. J. Math. Soc. Jpn. 56, 585–626 (2004)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Nishihara, K.: Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping. J. Differ. Equ. 137, 384–395 (1997)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Nishihara, K.: \(L^p-L^q\) estimates of solutions to the damped wave equation in 3-dimensional space and their application. Math. Z. 244, 631–649 (2003)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Nishihara, K.: \(L^p\)-\(L^q\) estimates for the 3-D damped wave equation and their application to the semilinear problem. Semin. Notes Math. Sci. 6, 69–83 (2003)Google Scholar
  48. 48.
    Nishihara, K.: Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term. Commun. Partial Differ. Equ. 35, 1402–1418 (2010)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Palmieri, A.: A global existence result for a semilinear wave equation with scale-invariant damping and mass in even space dimension, arXiv:1804.03978v1
  50. 50.
    Radu, P., Todorova, G., Yordanov, B.: Higher order energy decay rates for damped wave equations with variable coefficients. Discrete Contin. Dyn. Syst. Ser. S. 2, 609–629 (2009)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Radu, P., Todorova, G., Yordanov, B.: Decay estimates for wave equations with variable coefficients. Trans. Am. Math. Soc. 362, 2279–2299 (2010)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Radu, P., Todorova, G., Yordanov, B.: The generalized diffusion phenomenon and applications. SIAM J. Math. Anal. 48, 174–203 (2016)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Rauch, J., Taylor, M.: Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24, 79–86 (1974)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Sakata, S., Wakasugi, Y.: Movement of time-delayed hot spots in Euclidean space. Math. Z 285, 1007–1040 (2017)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Sobajima, M., Wakasugi, Y.: Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain. J. Differ. Equ. 261, 5690–5718 (2016)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Sobajima, M., Wakasugi, Y.: Remarks on an elliptic problem arising in weighted energy estimates for wave equations with space-dependent damping term in an exterior domain. AIMS Math. 2, 1–15 (2017)CrossRefGoogle Scholar
  57. 57.
    Sobajima, M., Wakasugi, Y.: Diffusion phenomena for the wave equation with space-dependent damping term growing at infinity. Adv. Differ. Equ. 23, 581–614 (2018)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Sobajima, M., Wakasugi, Y.: Weighted energy estimates for wave equation with space-dependent damping term for slowly decaying initial data, to appear in Commun. Contemp. Math., arXiv:1706.08311v1
  59. 59.
    Todorova, G., Yordanov, B.: Critical exponent for a nonlinear wave equation with damping. J. Differ. Equ. 174, 464–489 (2001)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Todorova, G., Yordanov, B.: Weighted \(L^2\)-estimates for dissipative wave equations with variable coefficients. J. Differ. Equ. 246, 4497–4518 (2009)CrossRefGoogle Scholar
  61. 61.
    Tu, Z., Lin, J.: A note on the blowup of scale invariant damping wave equation with sub-Strauss exponent, arXiv:1709.00866v2
  62. 62.
    Wakasa, K.: The lifespan of solutions to semilinear damped wave equations in one space dimension. Commun. Pure Appl. Anal. 15, 1265–1283 (2016)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Wakasa, K., Yordanov, B.: On the blow-up for critical semilinear wave equation with damping in the scattering case, arXiv:1807.06164v1
  64. 64.
    Wakasugi, Y.: Small data global existence for the semilinear wave equation with space–time dependent damping. J. Math. Anal. Appl. 393, 66–79 (2012)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Wakasugi, Y.: Critical exponent for the semilinear wave equation with scale invariant damping. In: Ruzhansky, M., Turunen, V. (eds.) Fourier Analysis, Trends in Mathematics, pp. 375–390. Birkhäuser, Basel (2014)CrossRefGoogle Scholar
  66. 66.
    Wakasugi, Y.: On diffusion phenomena for the linear wave equation with space-dependent damping. J. Hyp. Differ. Equ. 11, 795–819 (2014)MathSciNetCrossRefGoogle Scholar
  67. 67.
    Wakasugi, Y.: Scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients. J. Math. Anal. Appl. 447, 452–487 (2017)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Wakasugi, Y.: Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete Contin. Dyn. Syst. 34, 3831–3846 (2014)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Wirth, J.: Solution representations for a wave equation with weak dissipation. Math. Methods Appl. Sci. 27, 101–124 (2004)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Wirth, J.: Wave equations with time-dependent dissipation I. Non-effective dissipation. J. Differ. Equ. 222, 487–514 (2006)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Wirth, J.: Wave equations with time-dependent dissipation II. Effective dissipation. J. Differ. Equ. 232, 74–103 (2007)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Wirth, J.: Scattering and modified scattering for abstract wave equations with time-dependent dissipation. Adv. Differ. Equ. 12, 1115–1133 (2007)MathSciNetzbMATHGoogle Scholar
  73. 73.
    Yamazaki, T.: Asymptotic behavior for abstract wave equations with decaying dissipation. Adv. Differ. Equ. 11, 419–456 (2006)MathSciNetzbMATHGoogle Scholar
  74. 74.
    Yang, H., Milani, A.: On the diffusion phenomenon of quasilinear hyperbolic waves. Bull. Sci. Math. 124, 415–433 (2000)MathSciNetCrossRefGoogle Scholar
  75. 75.
    Zhang, Qi S.: A blow-up result for a nonlinear wave equation with damping: the critical case. C. R. Acad. Sci. Paris Sér. I Math. 333, 109–114 (2001)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Kenji Nishihara
    • 1
  • Motohiro Sobajima
    • 2
  • Yuta Wakasugi
    • 3
  1. 1.Waseda UniversityTokyoJapan
  2. 2.Department of Mathematics, Faculty of Science and TechnologyTokyo University of ScienceNoda-shiJapan
  3. 3.Department of Engineering for Production and Environment, Graduate School of Science and EngineeringEhime UniversityMatsuyamaJapan

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