June  2013, 2(2): 319-335. doi: 10.3934/eect.2013.2.319

Global existence for exterior problems of semilinear wave equations with the null condition in $2$D

1. 

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

Received  October 2012 Revised  February 2013 Published  March 2013

In this paper we deal with the exterior problem for a system of nonlinear wave equations in two space dimensions under some geometric restriction on the obstacle. We prove a global existence result for the problem with small and smooth initial data, provided that the nonlinearity is taken to be cubic and satisfies the null condition.
Citation: Hideo Kubo. Global existence for exterior problems of semilinear wave equations with the null condition in $2$D. Evolution Equations & Control Theory, 2013, 2 (2) : 319-335. doi: 10.3934/eect.2013.2.319
References:
[1]

S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I,, Invent. Math., 145 (2001), 597. doi: 10.1007/s002220100165.

[2]

P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions,, Comm. Partial Differential Equations, 18 (1993), 895. doi: 10.1080/03605309308820955.

[3]

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

[4]

A. Hoshiga and H. Kubo, Global solvability for systems of nonlinear wave equations with multiple speeds in two space dimensions,, Diff. Integral Eqs., 17 (2004), 593.

[5]

M. Ikawa, Mixed problems for hyperbolic equations of second order,, J. Math. Soc. Japan, 20 (1968), 580.

[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.

[7]

S. Katayama and H. Kubo, Decay estimates of a tangential derivative to the light cone for the wave equation and their application,, SIAM J. Math. Anal., 39 (2008), 1851. doi: 10.1137/070694417.

[8]

S. Katayama, H. Kubo and S. Lucente, Almost global existence for exterior Neumann problems of semilinear wave equations in 2D,, to appear in Commun. Pure Appl. Anal., ().

[9]

M. Keel, H. Smith and C. D. Sogge, On global existence for nonlinear wave equations outside of convex obstacles,, Amer. J. Math., 122 (2000), 805.

[10]

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. doi: 10.1006/jfan.2001.3844.

[11]

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

[12]

S. Klainerman, The null condition and global existence to nonlinear wave equations,, in, 23 (1986), 293.

[13]

H. Kubo, Uniform decay estimates for the wave equation in an exterior domain,, in, 47-1 (2007), 47.

[14]

H. Kubo, Almost global existence for nonlinear wave equations in an exterior domain in two space dimensions,, preprint, ().

[15]

K. Kubota, Existence of a global solutions to a semi-linear wave equation with initial data of noncompact support in low space dimensions,, Hokkaido Math. J., 22 (1993), 123.

[16]

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

[17]

J. Metcalfe, M. Nakamura and C. D. Sogge, Global existence of solutions to multiple speed systems of quasilinear wave equations in exterior domains,, Forum Math., 17 (2005), 133. doi: 10.1515/form.2005.17.1.133.

[18]

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.

[19]

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

[20]

J. Metcalfe and C. D. Sogge, Global existence of null-form wave equations in exterior domains,, Math. Z., 256 (2007), 521. doi: 10.1007/s00209-006-0083-2.

[21]

C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation,, Comm. Pure Appl. Math., 28 (1975), 229.

[22]

P. Secchi and Y. Shibata, On the decay of solutions to the 2D Neumann exterior problem for the wave equation,, J. Differential Equations, 194 (2003), 221. doi: 10.1016/S0022-0396(03)00189-X.

[23]

Y. Shibata and G. Nakamura, On a local existence theorem of Neumann problem for some quasilinear hyperbolic systems of 2nd order,, Math. Z., 202 (1989), 1. doi: 10.1007/BF01180683.

[24]

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. doi: 10.1007/BF01164023.

[25]

B. R. Vainberg, The short-wave asymptotic behavior of the solutions of stationary problems, and the asymptotic behavior as $t\rightarrow \infty $ of the solutions of nonstationary problems,, (Russian) Uspehi Mat. Nauk, 30 (1975), 3.

show all references

References:
[1]

S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I,, Invent. Math., 145 (2001), 597. doi: 10.1007/s002220100165.

[2]

P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions,, Comm. Partial Differential Equations, 18 (1993), 895. doi: 10.1080/03605309308820955.

[3]

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

[4]

A. Hoshiga and H. Kubo, Global solvability for systems of nonlinear wave equations with multiple speeds in two space dimensions,, Diff. Integral Eqs., 17 (2004), 593.

[5]

M. Ikawa, Mixed problems for hyperbolic equations of second order,, J. Math. Soc. Japan, 20 (1968), 580.

[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.

[7]

S. Katayama and H. Kubo, Decay estimates of a tangential derivative to the light cone for the wave equation and their application,, SIAM J. Math. Anal., 39 (2008), 1851. doi: 10.1137/070694417.

[8]

S. Katayama, H. Kubo and S. Lucente, Almost global existence for exterior Neumann problems of semilinear wave equations in 2D,, to appear in Commun. Pure Appl. Anal., ().

[9]

M. Keel, H. Smith and C. D. Sogge, On global existence for nonlinear wave equations outside of convex obstacles,, Amer. J. Math., 122 (2000), 805.

[10]

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. doi: 10.1006/jfan.2001.3844.

[11]

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

[12]

S. Klainerman, The null condition and global existence to nonlinear wave equations,, in, 23 (1986), 293.

[13]

H. Kubo, Uniform decay estimates for the wave equation in an exterior domain,, in, 47-1 (2007), 47.

[14]

H. Kubo, Almost global existence for nonlinear wave equations in an exterior domain in two space dimensions,, preprint, ().

[15]

K. Kubota, Existence of a global solutions to a semi-linear wave equation with initial data of noncompact support in low space dimensions,, Hokkaido Math. J., 22 (1993), 123.

[16]

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

[17]

J. Metcalfe, M. Nakamura and C. D. Sogge, Global existence of solutions to multiple speed systems of quasilinear wave equations in exterior domains,, Forum Math., 17 (2005), 133. doi: 10.1515/form.2005.17.1.133.

[18]

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.

[19]

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

[20]

J. Metcalfe and C. D. Sogge, Global existence of null-form wave equations in exterior domains,, Math. Z., 256 (2007), 521. doi: 10.1007/s00209-006-0083-2.

[21]

C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation,, Comm. Pure Appl. Math., 28 (1975), 229.

[22]

P. Secchi and Y. Shibata, On the decay of solutions to the 2D Neumann exterior problem for the wave equation,, J. Differential Equations, 194 (2003), 221. doi: 10.1016/S0022-0396(03)00189-X.

[23]

Y. Shibata and G. Nakamura, On a local existence theorem of Neumann problem for some quasilinear hyperbolic systems of 2nd order,, Math. Z., 202 (1989), 1. doi: 10.1007/BF01180683.

[24]

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. doi: 10.1007/BF01164023.

[25]

B. R. Vainberg, The short-wave asymptotic behavior of the solutions of stationary problems, and the asymptotic behavior as $t\rightarrow \infty $ of the solutions of nonstationary problems,, (Russian) Uspehi Mat. Nauk, 30 (1975), 3.

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