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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

Hideo Kubo 1,

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.
Keywords: pointwise decay estimate, Wave equation, exterior problem..
Mathematics Subject Classification: Primary: 35L15; Secondary: 35L2.
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, J. Math. Anal. Appl., 281 (2003), 22-45.  Google Scholar

[2]

P. Godin, Long time behavior of solutions to some nonlinear invariant mixed problems, Comm. Partial Differential Equations, 14 (1989), 299-374. doi: 10.1080/03605308908820599.  Google Scholar

[3]

P. Godin, Global existence of solutions to some exterior radial quasilinear Cauchy-Dirichlet problems, Amer. J. Math., 117 (1995), 1475-1505. doi: 10.2307/2375027.  Google Scholar

[4]

N. Hayashi, Global existence of small solutions to quadratic nonlinear wave equations in an exterior domain, J. Funct. Anal., 131 (1995), 302-344. doi: 10.1006/jfan.1995.1091.  Google Scholar

[5]

F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268. 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, J. Math. Soc. Japan, 60 (2008), 1135-1170.  Google Scholar

[7]

S. Katayama, H. Kubo, and S. Lucente, Almost global existence for exterior Neumann problems of semilinear wave equations in 2D, Commun. Pure Appl. Anal., 12 (2013), 2331-2360. 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, J. Funct. Anal., 189 (2002), 155-226. 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, J. Amer. Math. Soc., 17 (2004), 109-153. 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, Comm. Pure Appl. Math., 38 (1985), 321-332. 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, J. Math. Soc. Japan, 53 (2001), 875-912. doi: 10.2969/jmsj/05340875.  Google Scholar

[12]

H. Kubo, Uniform decay estimates for the wave equation in an exterior domain, in Asymptotic analysis and singularities, 31-54, Advanced Studies in Pure Mathematics 47-1, Math. Soc. of Japan, 2007.  Google Scholar

[13]

H. Kubo, Global existence for nonlinear wave equations in an exterior domain in 2D, Evolution Equations and Control Theory, 2 (2013), 319-335. 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, J. Differential Equations, 257 (2014), 2765-2800. ArXiv: 1204.3725v2. 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, Hokkaido Math. J., 22 (1993), 123-180. doi: 10.14492/hokmj/1381413170.  Google Scholar

[16]

J. Metcalfe, Global existence for semilinear wave equations exterior to nontrapping obstacles, Houston J. Math., 30 (2004), 259-281.  Google Scholar

[17]

J. Metcalfe, M. Nakamura and C. D. Sogge, Global existence of quasilinear, nonrelativistic wave equations satisfying the null condition, Japan. J. Math. (N.S.), 31 (2005), 391-472.  Google Scholar

[18]

J. Metcalfe and C. D. Sogge, Hyperbolic trapped rays and global existence of quasilinear wave equations, Invent. Math., 159 (2005), 75-117. 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, Math. Z., 191 (1986), 165-199. 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, J. Math. Anal. Appl., 281 (2003), 22-45.  Google Scholar

[2]

P. Godin, Long time behavior of solutions to some nonlinear invariant mixed problems, Comm. Partial Differential Equations, 14 (1989), 299-374. doi: 10.1080/03605308908820599.  Google Scholar

[3]

P. Godin, Global existence of solutions to some exterior radial quasilinear Cauchy-Dirichlet problems, Amer. J. Math., 117 (1995), 1475-1505. doi: 10.2307/2375027.  Google Scholar

[4]

N. Hayashi, Global existence of small solutions to quadratic nonlinear wave equations in an exterior domain, J. Funct. Anal., 131 (1995), 302-344. doi: 10.1006/jfan.1995.1091.  Google Scholar

[5]

F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268. 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, J. Math. Soc. Japan, 60 (2008), 1135-1170.  Google Scholar

[7]

S. Katayama, H. Kubo, and S. Lucente, Almost global existence for exterior Neumann problems of semilinear wave equations in 2D, Commun. Pure Appl. Anal., 12 (2013), 2331-2360. 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, J. Funct. Anal., 189 (2002), 155-226. 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, J. Amer. Math. Soc., 17 (2004), 109-153. 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, Comm. Pure Appl. Math., 38 (1985), 321-332. 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, J. Math. Soc. Japan, 53 (2001), 875-912. doi: 10.2969/jmsj/05340875.  Google Scholar

[12]

H. Kubo, Uniform decay estimates for the wave equation in an exterior domain, in Asymptotic analysis and singularities, 31-54, Advanced Studies in Pure Mathematics 47-1, Math. Soc. of Japan, 2007.  Google Scholar

[13]

H. Kubo, Global existence for nonlinear wave equations in an exterior domain in 2D, Evolution Equations and Control Theory, 2 (2013), 319-335. 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, J. Differential Equations, 257 (2014), 2765-2800. ArXiv: 1204.3725v2. 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, Hokkaido Math. J., 22 (1993), 123-180. doi: 10.14492/hokmj/1381413170.  Google Scholar

[16]

J. Metcalfe, Global existence for semilinear wave equations exterior to nontrapping obstacles, Houston J. Math., 30 (2004), 259-281.  Google Scholar

[17]

J. Metcalfe, M. Nakamura and C. D. Sogge, Global existence of quasilinear, nonrelativistic wave equations satisfying the null condition, Japan. J. Math. (N.S.), 31 (2005), 391-472.  Google Scholar

[18]

J. Metcalfe and C. D. Sogge, Hyperbolic trapped rays and global existence of quasilinear wave equations, Invent. Math., 159 (2005), 75-117. 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, Math. Z., 191 (1986), 165-199. doi: 10.1007/BF01164023.  Google Scholar

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