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 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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