July  2015, 14(4): 1469-1480. doi: 10.3934/cpaa.2015.14.1469

On the pointwise decay estimate for the wave equation with compactly supported forcing term

1. 

Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810

Received  September 2013 Revised  December 2013 Published  April 2015

In this paper we derive a new type of pointwise decay estimates for solutions to the Cauchy problem for the wave equation in 2D, in the sense that one can diminish the weight in the time variable for the forcing term if it is compactly supported in the spatial variables. As an application of the estimate, we also establish an improved decay estimate for the solution to the exterior problem in 2D.
Citation: Hideo Kubo. On the pointwise decay estimate for the wave equation with compactly supported forcing term. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1469-1480. doi: 10.3934/cpaa.2015.14.1469
References:
[1]

M. Di Flaviano, Lower bounds of the life span of classical solutions to a system of semilinear wave equations in two space dimensions,, \emph{J. Math. Anal. Appl.}, 281 (2003), 22.   Google Scholar

[2]

P. Godin, Long time behavior of solutions to some nonlinear invariant mixed problems,, \emph{Comm. Partial Differential Equations}, 14 (1989), 299.  doi: 10.1080/03605308908820599.  Google Scholar

[3]

P. Godin, Global existence of solutions to some exterior radial quasilinear Cauchy-Dirichlet problems,, \emph{Amer. J. Math.}, 117 (1995), 1475.  doi: 10.2307/2375027.  Google Scholar

[4]

N. Hayashi, Global existence of small solutions to quadratic nonlinear wave equations in an exterior domain,, \emph{J. Funct. Anal.}, 131 (1995), 302.  doi: 10.1006/jfan.1995.1091.  Google Scholar

[5]

F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions,, \emph{Manuscripta Math.}, 28 (1979), 235.  doi: 10.1007/BF01647974.  Google Scholar

[6]

S. Katayama and H. Kubo, An alternative proof of global existence for nonlinear wave equations in an exterior domain,, \emph{J. Math. Soc. Japan}, 60 (2008), 1135.   Google Scholar

[7]

S. Katayama, H. Kubo, and S. Lucente, Almost global existence for exterior Neumann problems of semilinear wave equations in 2D,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 2331.  doi: 10.3934/cpaa.2013.12.2331.  Google Scholar

[8]

M. Keel, H. Smith and C. D. Sogge, Global existence for a quasilinear wave equation outside of star-shaped domains,, \emph{J. Funct. Anal.}, 189 (2002), 155.  doi: 10.1006/jfan.2001.3844.  Google Scholar

[9]

M. Keel, H. Smith and C. D. Sogge, Almost global existence for quasilinear wave equations in three space dimensions,, \emph{J. Amer. Math. Soc.}, 17 (2004), 109.  doi: 10.1090/S0894-0347-03-00443-0.  Google Scholar

[10]

S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation,, \emph{Comm. Pure Appl. Math.}, 38 (1985), 321.  doi: 10.1002/cpa.3160380305.  Google Scholar

[11]

H. Kubo and K. Kubota, Asymptotic behavior of classical solutions to a system of semilinier wave equations in low space dimensions,, \emph{J. Math. Soc. Japan}, 53 (2001), 875.  doi: 10.2969/jmsj/05340875.  Google Scholar

[12]

H. Kubo, Uniform decay estimates for the wave equation in an exterior domain,, in \emph{Asymptotic analysis and singularities}, (2007), 31.   Google Scholar

[13]

H. Kubo, Global existence for nonlinear wave equations in an exterior domain in 2D,, \emph{Evolution Equations and Control Theory}, 2 (2013), 319.  doi: 10.3934/eect.2013.2.319.  Google Scholar

[14]

H. Kubo, Almost global existence for nonlinear wave equations in an exterior domain in 2D,, \emph{J. Differential Equations}, 257 (2014), 2765.  doi: 10.1016/j.jde.2014.05.048.  Google Scholar

[15]

K. Kubota, Existence of a global solutions to a semi-linear wave equation with initial data of non-compact support in low space dimensions,, \emph{Hokkaido Math. J.}, 22 (1993), 123.  doi: 10.14492/hokmj/1381413170.  Google Scholar

[16]

J. Metcalfe, Global existence for semilinear wave equations exterior to nontrapping obstacles,, \emph{Houston J. Math.}, 30 (2004), 259.   Google Scholar

[17]

J. Metcalfe, M. Nakamura and C. D. Sogge, Global existence of quasilinear, nonrelativistic wave equations satisfying the null condition,, \emph{Japan. J. Math. (N.S.)}, 31 (2005), 391.   Google Scholar

[18]

J. Metcalfe and C. D. Sogge, Hyperbolic trapped rays and global existence of quasilinear wave equations,, \emph{Invent. Math.}, 159 (2005), 75.  doi: 10.1007/s00222-004-0383-2.  Google Scholar

[19]

Y. Shibata and Y. Tsutsumi, On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain,, \emph{Math. Z.}, 191 (1986), 165.  doi: 10.1007/BF01164023.  Google Scholar

show all references

References:
[1]

M. Di Flaviano, Lower bounds of the life span of classical solutions to a system of semilinear wave equations in two space dimensions,, \emph{J. Math. Anal. Appl.}, 281 (2003), 22.   Google Scholar

[2]

P. Godin, Long time behavior of solutions to some nonlinear invariant mixed problems,, \emph{Comm. Partial Differential Equations}, 14 (1989), 299.  doi: 10.1080/03605308908820599.  Google Scholar

[3]

P. Godin, Global existence of solutions to some exterior radial quasilinear Cauchy-Dirichlet problems,, \emph{Amer. J. Math.}, 117 (1995), 1475.  doi: 10.2307/2375027.  Google Scholar

[4]

N. Hayashi, Global existence of small solutions to quadratic nonlinear wave equations in an exterior domain,, \emph{J. Funct. Anal.}, 131 (1995), 302.  doi: 10.1006/jfan.1995.1091.  Google Scholar

[5]

F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions,, \emph{Manuscripta Math.}, 28 (1979), 235.  doi: 10.1007/BF01647974.  Google Scholar

[6]

S. Katayama and H. Kubo, An alternative proof of global existence for nonlinear wave equations in an exterior domain,, \emph{J. Math. Soc. Japan}, 60 (2008), 1135.   Google Scholar

[7]

S. Katayama, H. Kubo, and S. Lucente, Almost global existence for exterior Neumann problems of semilinear wave equations in 2D,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 2331.  doi: 10.3934/cpaa.2013.12.2331.  Google Scholar

[8]

M. Keel, H. Smith and C. D. Sogge, Global existence for a quasilinear wave equation outside of star-shaped domains,, \emph{J. Funct. Anal.}, 189 (2002), 155.  doi: 10.1006/jfan.2001.3844.  Google Scholar

[9]

M. Keel, H. Smith and C. D. Sogge, Almost global existence for quasilinear wave equations in three space dimensions,, \emph{J. Amer. Math. Soc.}, 17 (2004), 109.  doi: 10.1090/S0894-0347-03-00443-0.  Google Scholar

[10]

S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation,, \emph{Comm. Pure Appl. Math.}, 38 (1985), 321.  doi: 10.1002/cpa.3160380305.  Google Scholar

[11]

H. Kubo and K. Kubota, Asymptotic behavior of classical solutions to a system of semilinier wave equations in low space dimensions,, \emph{J. Math. Soc. Japan}, 53 (2001), 875.  doi: 10.2969/jmsj/05340875.  Google Scholar

[12]

H. Kubo, Uniform decay estimates for the wave equation in an exterior domain,, in \emph{Asymptotic analysis and singularities}, (2007), 31.   Google Scholar

[13]

H. Kubo, Global existence for nonlinear wave equations in an exterior domain in 2D,, \emph{Evolution Equations and Control Theory}, 2 (2013), 319.  doi: 10.3934/eect.2013.2.319.  Google Scholar

[14]

H. Kubo, Almost global existence for nonlinear wave equations in an exterior domain in 2D,, \emph{J. Differential Equations}, 257 (2014), 2765.  doi: 10.1016/j.jde.2014.05.048.  Google Scholar

[15]

K. Kubota, Existence of a global solutions to a semi-linear wave equation with initial data of non-compact support in low space dimensions,, \emph{Hokkaido Math. J.}, 22 (1993), 123.  doi: 10.14492/hokmj/1381413170.  Google Scholar

[16]

J. Metcalfe, Global existence for semilinear wave equations exterior to nontrapping obstacles,, \emph{Houston J. Math.}, 30 (2004), 259.   Google Scholar

[17]

J. Metcalfe, M. Nakamura and C. D. Sogge, Global existence of quasilinear, nonrelativistic wave equations satisfying the null condition,, \emph{Japan. J. Math. (N.S.)}, 31 (2005), 391.   Google Scholar

[18]

J. Metcalfe and C. D. Sogge, Hyperbolic trapped rays and global existence of quasilinear wave equations,, \emph{Invent. Math.}, 159 (2005), 75.  doi: 10.1007/s00222-004-0383-2.  Google Scholar

[19]

Y. Shibata and Y. Tsutsumi, On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain,, \emph{Math. Z.}, 191 (1986), 165.  doi: 10.1007/BF01164023.  Google Scholar

[1]

Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37

[2]

Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307

[3]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064

[4]

Jeong Ja Bae, Mitsuhiro Nakao. Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 731-743. doi: 10.3934/dcds.2004.11.731

[5]

Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319

[6]

Neal Bez, Chris Jeavons. A sharp Sobolev-Strichartz estimate for the wave equation. Electronic Research Announcements, 2015, 22: 46-54. doi: 10.3934/era.2015.22.46

[7]

Seiji Ukai, Tong Yang, Huijiang Zhao. Exterior Problem of Boltzmann Equation with Temperature Difference. Communications on Pure & Applied Analysis, 2009, 8 (1) : 473-491. doi: 10.3934/cpaa.2009.8.473

[8]

Yongqin Liu, Weike Wang. The pointwise estimates of solutions for dissipative wave equation in multi-dimensions. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1013-1028. doi: 10.3934/dcds.2008.20.1013

[9]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387

[10]

Boris P. Belinskiy, Peter Caithamer. Energy estimate for the wave equation driven by a fractional Gaussian noise. Conference Publications, 2007, 2007 (Special) : 92-101. doi: 10.3934/proc.2007.2007.92

[11]

Li-Ming Yeh. Pointwise estimate for elliptic equations in periodic perforated domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1961-1986. doi: 10.3934/cpaa.2015.14.1961

[12]

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

[13]

Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559

[14]

Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations & Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017

[15]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367

[16]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095

[17]

Xue Yang, Xinglong Wu. Wave breaking and persistent decay of solution to a shallow water wave equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2149-2165. doi: 10.3934/dcdss.2016089

[18]

Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557

[19]

Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225

[20]

Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]