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

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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

Soichiro Katayama ^1, , Hideo Kubo ^2, and Sandra Lucente ^3,

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

Abstract
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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.

Keywords: almost global existence., Exterior Neumann problems, semilinear wave equations.

Mathematics Subject Classification: Primary: 35L20; Secondary: 35L7.

Citation: Soichiro Katayama, Hideo Kubo, Sandra Lucente. Almost global existence for exterior Neumann problems of semilinear wave equations in $2$D. Communications on Pure and 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), 623-737. doi: 10.1002/cpa.3160120405.
[2]	V. Georgiev and S. Lucente, Decay for Nonlinear Klein-Gordon Equations, NoDEA, 11 (2004), 529-555. doi: 10.1007/s00030-004-2027-z.
[3]	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.
[4]	M. Ikawa, Mixed problems for hyperbolic equations of second order, J. Math. Soc. Japan, 20 (1968), 580-608. doi: 10.2969/jmsj/02040580.
[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), 1135-1170. doi: 10.2969/jmsj/06041135.
[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, ().
[7]	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.
[8]	H. Kubo, Uniform decay estimates for the wave equation in an exterior domain, in "Asymptotic Analysis and Singularities," Advanced Studies in Pure Mathematics 47-1, Math. Soc. of Japan, (2007), 31-54.
[9]	H. Kubo, Global existence for nonlinear wave equations in an exterior domain in 2D, preprint,, \arXiv{1204.3725v2}., ().
[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), 123-180.
[11]	C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation, Comm. Pure Appl. Math., 28 (1975), 229-264. doi: 10.1002/cpa.3160280204.
[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), 221-236. doi: 10.1016/S0022-0396(03)00189-X.
[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), 1-64. doi: 10.1007/BF01180683.
[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), 165-199. doi: 10.1007/BF01164023.
[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), 3329-3347. doi: 10.1090/S0002-9947-2012-05607-8.
[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), 3-55 (Russian).
[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), 585-601. doi: 10.1016/j.jmaa.2010.08.052.
[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), 623-737.
[19]	V. Georgiev and S. Lucente, Decay for Nonlinear Klein-Gordon Equations, NoDEA, 11 (2004), 529-555.
[20]	P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations, 18 (1993), 895-916.
[21]	M. Ikawa, Mixed problems for hyperbolic equations of second order, J. Math. Soc. Japan, 20 (1968), 580-608.
[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), 1135-1170.
[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, ().
[24]	S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math., 38 (1985), 321-332.
[25]	H. Kubo, Uniform decay estimates for the wave equation in an exterior domain, in "Asymptotic Analysis and Singularities," Advanced Studies in Pure Mathematics 47-1, Math. Soc. of Japan, (2007), 31-54.
[26]	H. Kubo, Global existence for nonlinear wave equations in an exterior domain in 2D, preprint,, \arXiv{1204.3725v2}., ().
[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), 123-180.
[28]	C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation, Comm. Pure Appl. Math., 28 (1975), 229-264.
[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), 221-236.
[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), 1-64.
[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), 165-199.
[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), 3329-3347.
[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), 3-55 (Russian).
[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), 585-601.

show all references

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), 623-737. doi: 10.1002/cpa.3160120405.
[2]	V. Georgiev and S. Lucente, Decay for Nonlinear Klein-Gordon Equations, NoDEA, 11 (2004), 529-555. doi: 10.1007/s00030-004-2027-z.
[3]	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.
[4]	M. Ikawa, Mixed problems for hyperbolic equations of second order, J. Math. Soc. Japan, 20 (1968), 580-608. doi: 10.2969/jmsj/02040580.
[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), 1135-1170. doi: 10.2969/jmsj/06041135.
[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, ().
[7]	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.
[8]	H. Kubo, Uniform decay estimates for the wave equation in an exterior domain, in "Asymptotic Analysis and Singularities," Advanced Studies in Pure Mathematics 47-1, Math. Soc. of Japan, (2007), 31-54.
[9]	H. Kubo, Global existence for nonlinear wave equations in an exterior domain in 2D, preprint,, \arXiv{1204.3725v2}., ().
[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), 123-180.
[11]	C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation, Comm. Pure Appl. Math., 28 (1975), 229-264. doi: 10.1002/cpa.3160280204.
[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), 221-236. doi: 10.1016/S0022-0396(03)00189-X.
[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), 1-64. doi: 10.1007/BF01180683.
[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), 165-199. doi: 10.1007/BF01164023.
[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), 3329-3347. doi: 10.1090/S0002-9947-2012-05607-8.
[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), 3-55 (Russian).
[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), 585-601. doi: 10.1016/j.jmaa.2010.08.052.
[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), 623-737.
[19]	V. Georgiev and S. Lucente, Decay for Nonlinear Klein-Gordon Equations, NoDEA, 11 (2004), 529-555.
[20]	P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations, 18 (1993), 895-916.
[21]	M. Ikawa, Mixed problems for hyperbolic equations of second order, J. Math. Soc. Japan, 20 (1968), 580-608.
[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), 1135-1170.
[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, ().
[24]	S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math., 38 (1985), 321-332.
[25]	H. Kubo, Uniform decay estimates for the wave equation in an exterior domain, in "Asymptotic Analysis and Singularities," Advanced Studies in Pure Mathematics 47-1, Math. Soc. of Japan, (2007), 31-54.
[26]	H. Kubo, Global existence for nonlinear wave equations in an exterior domain in 2D, preprint,, \arXiv{1204.3725v2}., ().
[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), 123-180.
[28]	C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation, Comm. Pure Appl. Math., 28 (1975), 229-264.
[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), 221-236.
[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), 1-64.
[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), 165-199.
[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), 3329-3347.
[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), 3-55 (Russian).
[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), 585-601.

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