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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

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