# American Institute of Mathematical Sciences

November  2013, 12(6): 2331-2360. doi: 10.3934/cpaa.2013.12.2331

## Almost global existence for exterior Neumann problems of semilinear wave equations in $2$D

 1 Department of Mathematics, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan 2 Division of Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan 3 Dipartimento di Matematica, Università degli Studi di Bari, Via Orabona 4, 70125 Bari

Received  August 2012 Revised  January 2013 Published  May 2013

The aim of this article is to prove an "almost" global existence result for some semilinear wave equations in the plane outside a bounded convex obstacle with the Neumann boundary condition.
Citation: Soichiro Katayama, Hideo Kubo, Sandra Lucente. Almost global existence for exterior Neumann problems of semilinear wave equations in $2$D. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2331-2360. doi: 10.3934/cpaa.2013.12.2331
##### References:
 [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I,, Comm. Pure Appl. Math., 12 (1959), 623. doi: 10.1002/cpa.3160120405. Google Scholar [2] V. Georgiev and S. Lucente, Decay for Nonlinear Klein-Gordon Equations,, NoDEA, 11 (2004), 529. doi: 10.1007/s00030-004-2027-z. Google Scholar [3] P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions,, Comm. Partial Differential Equations, 18 (1993), 895. doi: 10.1080/03605309308820955. Google Scholar [4] M. Ikawa, Mixed problems for hyperbolic equations of second order,, J. Math. Soc. Japan, 20 (1968), 580. doi: 10.2969/jmsj/02040580. Google Scholar [5] 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. doi: 10.2969/jmsj/06041135. Google Scholar [6] S. Katayama and H. Kubo, Lower bound of the lifespan of solutions to semilinear wave equations in an exterior domain,, to appear in J. Hyper. Differential Equations, (). Google Scholar [7] 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. Google Scholar [8] H. Kubo, Uniform decay estimates for the wave equation in an exterior domain,, in, (2007), 47. Google Scholar [9] H. Kubo, Global existence for nonlinear wave equations in an exterior domain in 2D, preprint,, \arXiv{1204.3725v2}., (). Google Scholar [10] 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. Google Scholar [11] C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation,, Comm. Pure Appl. Math., 28 (1975), 229. doi: 10.1002/cpa.3160280204. Google Scholar [12] 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. Google Scholar [13] 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. Google Scholar [14] 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. Google Scholar [15] H. F. Smith, C. D. Sogge and C. Wang, Strichartz estimates for Dirichlet-Wave equations in two dimensions with applications,, Transactions Amer. Math. Soc., 364 (2012), 3329. doi: 10.1090/S0002-9947-2012-05607-8. Google Scholar [16] 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,, Uspehi Mat. Nauk, 30 (1975), 3. Google Scholar [17] Y. Zhou and W. Han, Blow-up of solutions to semilinear wave equations with variable coefficients and boundary,, J. Math. Anal. Appl., 374 (2011), 585. doi: 10.1016/j.jmaa.2010.08.052. Google Scholar [18] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I,, Comm. Pure Appl. Math., 12 (1959), 623. Google Scholar [19] V. Georgiev and S. Lucente, Decay for Nonlinear Klein-Gordon Equations,, NoDEA, 11 (2004), 529. Google Scholar [20] P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions,, Comm. Partial Differential Equations, 18 (1993), 895. Google Scholar [21] M. Ikawa, Mixed problems for hyperbolic equations of second order,, J. Math. Soc. Japan, 20 (1968), 580. Google Scholar [22] 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. Google Scholar [23] S. Katayama and H. Kubo, Lower bound of the lifespan of solutions to semilinear wave equations in an exterior domain,, to appear in J. Hyper. Differential Equations, (). Google Scholar [24] S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation,, Comm. Pure Appl. Math., 38 (1985), 321. Google Scholar [25] H. Kubo, Uniform decay estimates for the wave equation in an exterior domain,, in, (2007), 47. Google Scholar [26] H. Kubo, Global existence for nonlinear wave equations in an exterior domain in 2D, preprint,, \arXiv{1204.3725v2}., (). Google Scholar [27] 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. Google Scholar [28] C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation,, Comm. Pure Appl. Math., 28 (1975), 229. Google Scholar [29] 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. Google Scholar [30] 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. Google Scholar [31] 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. Google Scholar [32] H. F. Smith, C. D. Sogge and C. Wang, Strichartz estimates for Dirichlet-Wave equations in two dimensions with applications,, Transactions Amer. Math. Soc., 364 (2012), 3329. Google Scholar [33] 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,, Uspehi Mat. Nauk, 30 (1975), 3. Google Scholar [34] Y. Zhou and W. Han, Blow-up of solutions to semilinear wave equations with variable coefficients and boundary,, J. Math. Anal. Appl., 374 (2011), 585. Google Scholar

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##### References:
 [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I,, Comm. Pure Appl. Math., 12 (1959), 623. doi: 10.1002/cpa.3160120405. Google Scholar [2] V. Georgiev and S. Lucente, Decay for Nonlinear Klein-Gordon Equations,, NoDEA, 11 (2004), 529. doi: 10.1007/s00030-004-2027-z. Google Scholar [3] P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions,, Comm. Partial Differential Equations, 18 (1993), 895. doi: 10.1080/03605309308820955. Google Scholar [4] M. Ikawa, Mixed problems for hyperbolic equations of second order,, J. Math. Soc. Japan, 20 (1968), 580. doi: 10.2969/jmsj/02040580. Google Scholar [5] 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. doi: 10.2969/jmsj/06041135. Google Scholar [6] S. Katayama and H. Kubo, Lower bound of the lifespan of solutions to semilinear wave equations in an exterior domain,, to appear in J. Hyper. Differential Equations, (). Google Scholar [7] 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. Google Scholar [8] H. Kubo, Uniform decay estimates for the wave equation in an exterior domain,, in, (2007), 47. Google Scholar [9] H. Kubo, Global existence for nonlinear wave equations in an exterior domain in 2D, preprint,, \arXiv{1204.3725v2}., (). Google Scholar [10] 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. Google Scholar [11] C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation,, Comm. Pure Appl. Math., 28 (1975), 229. doi: 10.1002/cpa.3160280204. Google Scholar [12] 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. Google Scholar [13] 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. Google Scholar [14] 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. Google Scholar [15] H. F. Smith, C. D. Sogge and C. Wang, Strichartz estimates for Dirichlet-Wave equations in two dimensions with applications,, Transactions Amer. Math. Soc., 364 (2012), 3329. doi: 10.1090/S0002-9947-2012-05607-8. Google Scholar [16] 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,, Uspehi Mat. Nauk, 30 (1975), 3. Google Scholar [17] Y. Zhou and W. Han, Blow-up of solutions to semilinear wave equations with variable coefficients and boundary,, J. Math. Anal. Appl., 374 (2011), 585. doi: 10.1016/j.jmaa.2010.08.052. Google Scholar [18] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I,, Comm. Pure Appl. Math., 12 (1959), 623. Google Scholar [19] V. Georgiev and S. Lucente, Decay for Nonlinear Klein-Gordon Equations,, NoDEA, 11 (2004), 529. Google Scholar [20] P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions,, Comm. Partial Differential Equations, 18 (1993), 895. Google Scholar [21] M. Ikawa, Mixed problems for hyperbolic equations of second order,, J. Math. Soc. Japan, 20 (1968), 580. Google Scholar [22] 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. Google Scholar [23] S. Katayama and H. Kubo, Lower bound of the lifespan of solutions to semilinear wave equations in an exterior domain,, to appear in J. Hyper. Differential Equations, (). Google Scholar [24] S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation,, Comm. Pure Appl. Math., 38 (1985), 321. Google Scholar [25] H. Kubo, Uniform decay estimates for the wave equation in an exterior domain,, in, (2007), 47. Google Scholar [26] H. Kubo, Global existence for nonlinear wave equations in an exterior domain in 2D, preprint,, \arXiv{1204.3725v2}., (). Google Scholar [27] 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. Google Scholar [28] C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation,, Comm. Pure Appl. Math., 28 (1975), 229. Google Scholar [29] 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. Google Scholar [30] 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. Google Scholar [31] 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. Google Scholar [32] H. F. Smith, C. D. Sogge and C. Wang, Strichartz estimates for Dirichlet-Wave equations in two dimensions with applications,, Transactions Amer. Math. Soc., 364 (2012), 3329. Google Scholar [33] 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,, Uspehi Mat. Nauk, 30 (1975), 3. Google Scholar [34] Y. Zhou and W. Han, Blow-up of solutions to semilinear wave equations with variable coefficients and boundary,, J. Math. Anal. Appl., 374 (2011), 585. Google Scholar
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