# American Institute of Mathematical Sciences

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 and 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), 597-618. doi: 10.1007/s002220100165. [2] P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations, 18 (1993), 895-916. doi: 10.1080/03605309308820955. [3] P. Godin, Global existence of solutions to some exterior radial quasilinear Cauchy-Dirichlet problems, Amer. J. Math., 117 (1995), 1475-1505. 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), 593-622. [5] M. Ikawa, Mixed problems for hyperbolic equations of second order, J. Math. Soc. Japan, 20 (1968), 580-608. [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 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), 1851-1862. 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), 805-842. [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), 155-226. 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), 321-332. doi: 10.1002/cpa.3160380305. [12] S. Klainerman, The null condition and global existence to nonlinear wave equations, in "Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1" (Santa Fe, N.M., 1984), Lectures in Appl. Math., 23, Amer. Math. Soc., Providence, RI, (1986), 293-326. [13] H. Kubo, Uniform decay estimates for the wave equation in an exterior domain, in "Asymptotic Analysis and Singularities—hyperbolic and Dispersive PDEs and Fluid Mechanics", Advanced Studies in Pure Mathematics, 47-1, Math. Soc. of Japan, Tokyo, (2007), 31-54. [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), 123-180. [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 solutions to multiple speed systems of quasilinear wave equations in exterior domains, Forum Math., 17 (2005), 133-168. 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), 391-472. [19] 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. [20] J. Metcalfe and C. D. Sogge, Global existence of null-form wave equations in exterior domains, Math. Z., 256 (2007), 521-549. 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), 229-264. [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), 221-236. 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), 1-64. 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), 165-199. 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), 3-55.

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##### References:
 [1] S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math., 145 (2001), 597-618. doi: 10.1007/s002220100165. [2] P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations, 18 (1993), 895-916. doi: 10.1080/03605309308820955. [3] P. Godin, Global existence of solutions to some exterior radial quasilinear Cauchy-Dirichlet problems, Amer. J. Math., 117 (1995), 1475-1505. 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), 593-622. [5] M. Ikawa, Mixed problems for hyperbolic equations of second order, J. Math. Soc. Japan, 20 (1968), 580-608. [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 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), 1851-1862. 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), 805-842. [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), 155-226. 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), 321-332. doi: 10.1002/cpa.3160380305. [12] S. Klainerman, The null condition and global existence to nonlinear wave equations, in "Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1" (Santa Fe, N.M., 1984), Lectures in Appl. Math., 23, Amer. Math. Soc., Providence, RI, (1986), 293-326. [13] H. Kubo, Uniform decay estimates for the wave equation in an exterior domain, in "Asymptotic Analysis and Singularities—hyperbolic and Dispersive PDEs and Fluid Mechanics", Advanced Studies in Pure Mathematics, 47-1, Math. Soc. of Japan, Tokyo, (2007), 31-54. [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), 123-180. [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 solutions to multiple speed systems of quasilinear wave equations in exterior domains, Forum Math., 17 (2005), 133-168. 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), 391-472. [19] 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. [20] J. Metcalfe and C. D. Sogge, Global existence of null-form wave equations in exterior domains, Math. Z., 256 (2007), 521-549. 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), 229-264. [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), 221-236. 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), 1-64. 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), 165-199. 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), 3-55.
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