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

Large time behavior of solutions for a nonlinear damped wave equation

Hiroshi Takeda 1,

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.
Keywords: hyperbolic equations, asymptotic profile, Damped wave equation, perturbation., Cauchy problem.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35B3.
Citation: Hiroshi Takeda. Large time behavior of solutions for a nonlinear damped wave equation. Communications on Pure & 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

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