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

Article

## 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 . 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

## References

1. 1.
Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weights. Compos. Math. 53, 259–275 (1984)
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)
3. 3.
D’Abbicco, M.: The threshold of effective damping for semilinear wave equations. Math. Methods Appl. Sci. 38, 1032–1045 (2015)
4. 4.
D’Abbicco, M., Lucente, S.: A modified test function method for damped wave equations. Adv. Nonlinear Stud. 13, 867–892 (2013)
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)
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)
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)
11. 11.
Hayashi, N., Kaikina, E.I., Naumkin, P.I.: Damped wave equation with super critical nonlinearities. Differ. Integral Equ. 17, 637–652 (2004)
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)
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)
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)
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)
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)
24. 24.
Ikehata, R., Todorova, G., Yordanov, B.: Critical exponent for semilinear wave equations with space-dependent potential. Funkcialaj Ekvacioj 52, 411–435 (2009)
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)
26. 26.
Karch, G.: Selfsimilar profiles in large time asymptotics of solutions to damped wave equations. Studia Math. 143, 175–197 (2000)
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)
28. 28.
Khader, M.: Nonlinear dissipative wave equations with space–time dependent potential. Nonlinear Anal. 74, 3945–3963 (2011)
29. 29.
Khader, M.: Global existence for the dissipative wave equations with space–time dependent potential. Nonlinear Anal. 81, 87–100 (2013)
30. 30.
Kirane, M., Qafsaoui, M.: Fujita’s exponent for a semilinear wave equation with linear damping. Adv. Nonlinear Stud. 2, 41–49 (2002)
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)
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)
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)
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)
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)
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)
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)
40. 40.
Matsumura, A.: On the asymptotic behavior of solutions of semi-linear wave equations. Publ. Res. Inst. Math. Sci. 12, 169–189 (1976)
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)
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)
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)
45. 45.
Nishihara, K.: Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping. J. Differ. Equ. 137, 384–395 (1997)
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)
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)
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)
51. 51.
Radu, P., Todorova, G., Yordanov, B.: Decay estimates for wave equations with variable coefficients. Trans. Am. Math. Soc. 362, 2279–2299 (2010)
52. 52.
Radu, P., Todorova, G., Yordanov, B.: The generalized diffusion phenomenon and applications. SIAM J. Math. Anal. 48, 174–203 (2016)
53. 53.
Rauch, J., Taylor, M.: Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24, 79–86 (1974)
54. 54.
Sakata, S., Wakasugi, Y.: Movement of time-delayed hot spots in Euclidean space. Math. Z 285, 1007–1040 (2017)
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)
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)
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)
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)
60. 60.
Todorova, G., Yordanov, B.: Weighted $$L^2$$-estimates for dissipative wave equations with variable coefficients. J. Differ. Equ. 246, 4497–4518 (2009)
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)
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)
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)
66. 66.
Wakasugi, Y.: On diffusion phenomena for the linear wave equation with space-dependent damping. J. Hyp. Differ. Equ. 11, 795–819 (2014)
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)
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)
69. 69.
Wirth, J.: Solution representations for a wave equation with weak dissipation. Math. Methods Appl. Sci. 27, 101–124 (2004)
70. 70.
Wirth, J.: Wave equations with time-dependent dissipation I. Non-effective dissipation. J. Differ. Equ. 222, 487–514 (2006)
71. 71.
Wirth, J.: Wave equations with time-dependent dissipation II. Effective dissipation. J. Differ. Equ. 232, 74–103 (2007)
72. 72.
Wirth, J.: Scattering and modified scattering for abstract wave equations with time-dependent dissipation. Adv. Differ. Equ. 12, 1115–1133 (2007)
73. 73.
Yamazaki, T.: Asymptotic behavior for abstract wave equations with decaying dissipation. Adv. Differ. Equ. 11, 419–456 (2006)
74. 74.
Yang, H., Milani, A.: On the diffusion phenomenon of quasilinear hyperbolic waves. Bull. Sci. Math. 124, 415–433 (2000)
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)

© Springer Nature Switzerland AG 2018

## Authors and Affiliations

• Kenji Nishihara
• 1
• Motohiro Sobajima
• 2
• Yuta Wakasugi
• 3
Email author
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