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), 597-618. doi: 10.1007/s002220100165.  Google Scholar

[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.  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]

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.  Google Scholar

[5]

M. Ikawa, Mixed problems for hyperbolic equations of second order, J. Math. Soc. Japan, 20 (1968), 580-608.  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 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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  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 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation, Comm. Pure Appl. Math., 28 (1975), 229-264.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  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]

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.  Google Scholar

[5]

M. Ikawa, Mixed problems for hyperbolic equations of second order, J. Math. Soc. Japan, 20 (1968), 580-608.  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 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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  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 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation, Comm. Pure Appl. Math., 28 (1975), 229-264.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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