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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 |
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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