# American Institute of Mathematical Sciences

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.
 [1] Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations and 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 and 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 and Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064 [4] Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n}$. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319 [5] Jeong Ja Bae, Mitsuhiro Nakao. Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 731-743. doi: 10.3934/dcds.2004.11.731 [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 and 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 and Continuous Dynamical Systems, 2008, 20 (4) : 1013-1028. doi: 10.3934/dcds.2008.20.1013 [9] Li-Ming Yeh. Pointwise estimate for elliptic equations in periodic perforated domains. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1961-1986. doi: 10.3934/cpaa.2015.14.1961 [10] Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387 [11] 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 [12] Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure and 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 and 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 and Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017 [15] Monica Conti, Lorenzo Liverani, Vittorino Pata. On the optimal decay rate of the weakly damped wave equation. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022107 [16] Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367 [17] Soumen Senapati, Manmohan Vashisth. Stability estimate for a partial data inverse problem for the convection-diffusion equation. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021060 [18] Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557 [19] Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095 [20] Xue Yang, Xinglong Wu. Wave breaking and persistent decay of solution to a shallow water wave equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2149-2165. doi: 10.3934/dcdss.2016089

2021 Impact Factor: 1.273