American Institute of Mathematical Sciences

January  2016, 15(1): 41-55. doi: 10.3934/cpaa.2016.15.41

Large time behavior of solutions for a nonlinear damped wave equation

 1 Fukuoka Institute of Technology, Wajiro-higashi, Higashi-ku, Fukuoka, 811-0295

Received  October 2014 Revised  September 2015 Published  December 2015

We study the large time behavior of small solutions to the Cauchy problem for a nonlinear damped wave equation. We proved that the solution is approximated by the Gauss kernel with suitable choice of the coefficients and powers of $t$ for $N+1$ th order for all $N \in \mathbb{N}$. Our analysis is based on the approximation theorem of the linear solution by the solution of the heat equation [37]. In particular, as pointed out by Galley-Raugel [4], we explicitly observe that from third order expansion, the asymptotic behavior of the solutions of a nonlinear damped wave equation is different from that of a nonlinear heat equation.
Citation: Hiroshi Takeda. Large time behavior of solutions for a nonlinear damped wave equation. Communications on Pure &amp; Applied Analysis, 2016, 15 (1) : 41-55. doi: 10.3934/cpaa.2016.15.41
References:
 [1] H. Bellout and A. Friedman, Blow-up estimates for a nonlinear hyperbolic heat equation, SIAM J. Math. Anal., 20 (1989), 354-366. doi: 10.1137/0520022.  Google Scholar [2] W. Dan and Y. Shibata, On a local energy decay of solutions of a dissipative wave equation, Funkcial. Ekvac., 38 (1995), 545-568.  Google Scholar [3] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 12 (1966), 109-124.  Google Scholar [4] Th. Gallay and G. Raugel, Scaling variables and asymptotic expansions in damped wave equations, J. Differential Equations, 150 (1998), 42-97. doi: 10.1006/jdeq.1998.3459.  Google Scholar [5] M-H. Giga, Y. Giga and J. Saal, Nonlinear partial differential equations, Asymptotic behavior of solutions and self-similar solutions, Progress in Nonlinear Differential Equations and their Applications, 79, Birkhäuser Boston, Inc., Boston, MA, 2010. doi: 10.1007/978-0-8176-4651-6.  Google Scholar [6] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503-505.  Google Scholar [7] N. Hayashi, E. Kaikina and PI Naumkin, Damped wave equation with super critical nonlinearities, Differential Integral Equations, 17 (2004), 637-652.  Google Scholar [8] N, Hayashi, E. Kaikina and PI Naumkin, Damped wave equation with a critical nonlinearity, Trans. Amer. Math. Soc., 358 (2006), 1165-1185. doi: 10.1090/S0002-9947-05-03818-3.  Google Scholar [9] N. Hayashi, E. Kaikina and PI Naumkin, On the critical nonlinear damped wave equation with large initial data, J. Math. Anal. Appl., 334 (2007), 1400-1425. doi: 10.1016/j.jmaa.2007.01.021.  Google Scholar [10] F. Hirosawa and J. Wirth, $C^m$-theory of damped wave equations with stabilisation, J. Math. Anal. Appl., 343 (2008), 1022-1035. doi: 10.1016/j.jmaa.2008.02.024.  Google Scholar [11] T. Hosono and T. Ogawa, Large time behavior and $L^p$-$L^q$ estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations, 203 (2004), 82-118. doi: 10.1016/j.jde.2004.03.034.  Google Scholar [12] K. Ishige and T. Kawakami, Refined asymptotic profiles for a semilinear heat equation, Math. Ann., 353 (2012), 161-192. doi: 10.1007/s00208-011-0677-9.  Google Scholar [13] R. Ikehata, T. Miyaoka and T. Nakatake, Decay estimates of solutions for dissipative wave equations in $\mathbbR^n$ with lower power nonlinearities, J. Math. Soc. Japan, 56 (2004), 365-373. doi: 10.2969/jmsj/1191418635.  Google Scholar [14] R. Ikehata and K. Tanizawa, Global existence of solutions for semilinear wave equations in $\mathbbR^N$ with non-compactly supported initial data, Nonlinear Anal. T.M.A., 61 (2005), 1189-1208. doi: 10.1016/j.na.2005.01.097.  Google Scholar [15] G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math., 143 (2000), 175-197.  Google Scholar [16] T. Kawakami and Y. Ueda, Asymptotic profiles to the solutions for a nonlinear damped wave equations, Differential Integral Equations, 26 (2013), 781-814.  Google Scholar [17] S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan, 47 (1995), 617-653. doi: 10.2969/jmsj/04740617.  Google Scholar [18] T-T. Li, Nonlinear heat conduction with finite speed of propagation, in Proceedings of the China-Japan Symposium on Reaction-Diffusion Equations and their Applications an Computational Aspect World Sci. Publ., River Edge, NJ, (1997), 81-91.  Google Scholar [19] T-T. Li and Y. Zhou, Breakdown of solutions to $\square u + u_t = | u|^{1 + \alpha}$, Discrete Contin. Dynam. Syst., 1 (1995), 503-520. doi: 10.3934/dcds.1995.1.503.  Google Scholar [20] P. Marcati and K. Nishihara, The $L^p$-$L^q$ estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations, 191 (2003), 445-469. doi: 10.1016/S0022-0396(03)00026-3.  Google Scholar [21] A. Matsumura, On the asymptotic behavior of solutions of semilinear wave equations, Publ. Res. Inst. Sci. Kyoto Univ., 12 (1976), 169-189.  Google Scholar [22] M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214 (1993), 325-342. doi: 10.1007/BF02572407.  Google Scholar [23] T. Narazaki, $L^p$-$L^q$ estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585-626. doi: 10.2969/jmsj/1191418647.  Google Scholar [24] T. Narazaki, Global solutions to the Cauchy problem for a system of damped wave equations, Differential Integral Equations, 24 (2011), 569-600.  Google Scholar [25] K. Nishihara, $L^p$-$L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z., 244 (2003), 631-649.  Google Scholar [26] K. Nishihara, Asymptotic behavior of solutions for a system of semilinear heat equations and the corresponding damped wave system, Osaka J. Math., 49 (2012), 331-348.  Google Scholar [27] K. Nishihara and H. Zhao, Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption, J. Math. Anal. Appl., 313 (2006), 598-610. doi: 10.1016/j.jmaa.2005.08.059.  Google Scholar [28] T. Ogawa and H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. T.M.A., 70 (2009), 3696-3701. doi: 10.1016/j.na.2008.07.025.  Google Scholar [29] T. Ogawa and H. Takeda, Global existence of solutions for a system of nonlinear damped wave equations, Differential Integral Equations, 23 (2010), 635-657.  Google Scholar [30] T. Ogawa and H. Takeda, Large time behavior of solutions for a system of nonlinear damped wave equations, J. Differential Equations, 251 (2011), 3090-3113. doi: 10.1016/j.jde.2011.07.034.  Google Scholar [31] K. Ono, Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations, Discrete Contin. Dynam. Systems, 9 (2003), 651-662. doi: 10.3934/dcds.2003.9.651.  Google Scholar [32] R. Orive, E. Zuazua and A. Pazoto, Asymptotic expansion for damped wave equations with periodic coefficients, Math. Models Methods Appl. Sci., 11 (2001), 1285-1310. doi: 10.1142/S0218202501001331.  Google Scholar [33] R. Racke, Decay rates for solutions of damped systems and generalized Fourier transforms, J. Reine Angew. Math., 412 (1990), 1-19. doi: 10.1515/crll.1990.412.1.  Google Scholar [34] P. Radu, G. Todorova and B. Yordanov, Higher order energy decay rates for damped wave equations with variable coefficients, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 609-629. doi: 10.3934/dcdss.2009.2.609.  Google Scholar [35] F. Sun and M. Wang, Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping, Nonlinear Anal. T.M.A., 66 (2007), 2889-2910. doi: 10.1016/j.na.2006.04.012.  Google Scholar [36] H. Takeda, Global existence and nonexistence of solutions for a system of nonlinear damped wave equations, J. Math. Anal. Appl., 360 (2009), 631-650. doi: 10.1016/j.jmaa.2009.06.072.  Google Scholar [37] H. Takeda, Higher-order expansion of solutions for a damped wave equation, Asymptot. Anal., 94 (2015), 1-31.  Google Scholar [38] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489. doi: 10.1006/jdeq.2000.3933.  Google Scholar [39] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.  Google Scholar [40] Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris Ser. I Math., 333 (2001), 109-114. doi: 10.1016/S0764-4442(01)01999-1.  Google Scholar

show all references

References:
 [1] H. Bellout and A. Friedman, Blow-up estimates for a nonlinear hyperbolic heat equation, SIAM J. Math. Anal., 20 (1989), 354-366. doi: 10.1137/0520022.  Google Scholar [2] W. Dan and Y. Shibata, On a local energy decay of solutions of a dissipative wave equation, Funkcial. Ekvac., 38 (1995), 545-568.  Google Scholar [3] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 12 (1966), 109-124.  Google Scholar [4] Th. Gallay and G. Raugel, Scaling variables and asymptotic expansions in damped wave equations, J. Differential Equations, 150 (1998), 42-97. doi: 10.1006/jdeq.1998.3459.  Google Scholar [5] M-H. Giga, Y. Giga and J. Saal, Nonlinear partial differential equations, Asymptotic behavior of solutions and self-similar solutions, Progress in Nonlinear Differential Equations and their Applications, 79, Birkhäuser Boston, Inc., Boston, MA, 2010. doi: 10.1007/978-0-8176-4651-6.  Google Scholar [6] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503-505.  Google Scholar [7] N. Hayashi, E. Kaikina and PI Naumkin, Damped wave equation with super critical nonlinearities, Differential Integral Equations, 17 (2004), 637-652.  Google Scholar [8] N, Hayashi, E. Kaikina and PI Naumkin, Damped wave equation with a critical nonlinearity, Trans. Amer. Math. Soc., 358 (2006), 1165-1185. doi: 10.1090/S0002-9947-05-03818-3.  Google Scholar [9] N. Hayashi, E. Kaikina and PI Naumkin, On the critical nonlinear damped wave equation with large initial data, J. Math. Anal. Appl., 334 (2007), 1400-1425. doi: 10.1016/j.jmaa.2007.01.021.  Google Scholar [10] F. Hirosawa and J. Wirth, $C^m$-theory of damped wave equations with stabilisation, J. Math. Anal. Appl., 343 (2008), 1022-1035. doi: 10.1016/j.jmaa.2008.02.024.  Google Scholar [11] T. Hosono and T. Ogawa, Large time behavior and $L^p$-$L^q$ estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations, 203 (2004), 82-118. doi: 10.1016/j.jde.2004.03.034.  Google Scholar [12] K. Ishige and T. Kawakami, Refined asymptotic profiles for a semilinear heat equation, Math. Ann., 353 (2012), 161-192. doi: 10.1007/s00208-011-0677-9.  Google Scholar [13] R. Ikehata, T. Miyaoka and T. Nakatake, Decay estimates of solutions for dissipative wave equations in $\mathbbR^n$ with lower power nonlinearities, J. Math. Soc. Japan, 56 (2004), 365-373. doi: 10.2969/jmsj/1191418635.  Google Scholar [14] R. Ikehata and K. Tanizawa, Global existence of solutions for semilinear wave equations in $\mathbbR^N$ with non-compactly supported initial data, Nonlinear Anal. T.M.A., 61 (2005), 1189-1208. doi: 10.1016/j.na.2005.01.097.  Google Scholar [15] G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math., 143 (2000), 175-197.  Google Scholar [16] T. Kawakami and Y. Ueda, Asymptotic profiles to the solutions for a nonlinear damped wave equations, Differential Integral Equations, 26 (2013), 781-814.  Google Scholar [17] S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan, 47 (1995), 617-653. doi: 10.2969/jmsj/04740617.  Google Scholar [18] T-T. Li, Nonlinear heat conduction with finite speed of propagation, in Proceedings of the China-Japan Symposium on Reaction-Diffusion Equations and their Applications an Computational Aspect World Sci. Publ., River Edge, NJ, (1997), 81-91.  Google Scholar [19] T-T. Li and Y. Zhou, Breakdown of solutions to $\square u + u_t = | u|^{1 + \alpha}$, Discrete Contin. Dynam. Syst., 1 (1995), 503-520. doi: 10.3934/dcds.1995.1.503.  Google Scholar [20] P. Marcati and K. Nishihara, The $L^p$-$L^q$ estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations, 191 (2003), 445-469. doi: 10.1016/S0022-0396(03)00026-3.  Google Scholar [21] A. Matsumura, On the asymptotic behavior of solutions of semilinear wave equations, Publ. Res. Inst. Sci. Kyoto Univ., 12 (1976), 169-189.  Google Scholar [22] M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214 (1993), 325-342. doi: 10.1007/BF02572407.  Google Scholar [23] T. Narazaki, $L^p$-$L^q$ estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585-626. doi: 10.2969/jmsj/1191418647.  Google Scholar [24] T. Narazaki, Global solutions to the Cauchy problem for a system of damped wave equations, Differential Integral Equations, 24 (2011), 569-600.  Google Scholar [25] K. Nishihara, $L^p$-$L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z., 244 (2003), 631-649.  Google Scholar [26] K. Nishihara, Asymptotic behavior of solutions for a system of semilinear heat equations and the corresponding damped wave system, Osaka J. Math., 49 (2012), 331-348.  Google Scholar [27] K. Nishihara and H. Zhao, Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption, J. Math. Anal. Appl., 313 (2006), 598-610. doi: 10.1016/j.jmaa.2005.08.059.  Google Scholar [28] T. Ogawa and H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. T.M.A., 70 (2009), 3696-3701. doi: 10.1016/j.na.2008.07.025.  Google Scholar [29] T. Ogawa and H. Takeda, Global existence of solutions for a system of nonlinear damped wave equations, Differential Integral Equations, 23 (2010), 635-657.  Google Scholar [30] T. Ogawa and H. Takeda, Large time behavior of solutions for a system of nonlinear damped wave equations, J. Differential Equations, 251 (2011), 3090-3113. doi: 10.1016/j.jde.2011.07.034.  Google Scholar [31] K. Ono, Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations, Discrete Contin. Dynam. Systems, 9 (2003), 651-662. doi: 10.3934/dcds.2003.9.651.  Google Scholar [32] R. Orive, E. Zuazua and A. Pazoto, Asymptotic expansion for damped wave equations with periodic coefficients, Math. Models Methods Appl. Sci., 11 (2001), 1285-1310. doi: 10.1142/S0218202501001331.  Google Scholar [33] R. Racke, Decay rates for solutions of damped systems and generalized Fourier transforms, J. Reine Angew. Math., 412 (1990), 1-19. doi: 10.1515/crll.1990.412.1.  Google Scholar [34] P. Radu, G. Todorova and B. Yordanov, Higher order energy decay rates for damped wave equations with variable coefficients, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 609-629. doi: 10.3934/dcdss.2009.2.609.  Google Scholar [35] F. Sun and M. Wang, Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping, Nonlinear Anal. T.M.A., 66 (2007), 2889-2910. doi: 10.1016/j.na.2006.04.012.  Google Scholar [36] H. Takeda, Global existence and nonexistence of solutions for a system of nonlinear damped wave equations, J. Math. Anal. Appl., 360 (2009), 631-650. doi: 10.1016/j.jmaa.2009.06.072.  Google Scholar [37] H. Takeda, Higher-order expansion of solutions for a damped wave equation, Asymptot. Anal., 94 (2015), 1-31.  Google Scholar [38] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489. doi: 10.1006/jdeq.2000.3933.  Google Scholar [39] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.  Google Scholar [40] Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris Ser. I Math., 333 (2001), 109-114. doi: 10.1016/S0764-4442(01)01999-1.  Google Scholar
 [1] Takashi Narazaki. Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations. Conference Publications, 2009, 2009 (Special) : 592-601. doi: 10.3934/proc.2009.2009.592 [2] Ryo Ikehata, Marina Soga. Asymptotic profiles for a strongly damped plate equation with lower order perturbation. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1759-1780. doi: 10.3934/cpaa.2015.14.1759 [3] Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401 [4] Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735 [5] Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure & Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921 [6] Sergey Zelik. Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 351-392. doi: 10.3934/dcds.2004.11.351 [7] M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215 [8] Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019 [9] V. Pata, Sergey Zelik. A remark on the damped wave equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 611-616. doi: 10.3934/cpaa.2006.5.611 [10] V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731 [11] Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012 [12] Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045 [13] Akisato Kubo. Asymptotic behavior of solutions of the mixed problem for semilinear hyperbolic equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 59-74. doi: 10.3934/cpaa.2004.3.59 [14] Todor Gramchev, Nicola Orrú. Cauchy problem for a class of nondiagonalizable hyperbolic systems. Conference Publications, 2011, 2011 (Special) : 533-542. doi: 10.3934/proc.2011.2011.533 [15] Lorena Bociu, Petronela Radu. Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping. Conference Publications, 2009, 2009 (Special) : 60-71. doi: 10.3934/proc.2009.2009.60 [16] Mohammad Kafini. On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021093 [17] Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048 [18] Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149 [19] Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1541-1551. doi: 10.3934/cpaa.2014.13.1541 [20] Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $L^p$-$L^q$ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090

2020 Impact Factor: 1.916