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

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. 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

[1]

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

[2]

Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651

[3]

Jason Metcalfe, Christopher D. Sogge. Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1589-1601. doi: 10.3934/dcds.2010.28.1589

[4]

Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018

[5]

John M. Ball. Global attractors for damped semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31

[6]

Jason Metcalfe, Jacob Perry. Global solutions to quasilinear wave equations in homogeneous waveguides with Neumann boundary conditions. Communications on Pure & Applied Analysis, 2012, 11 (2) : 547-556. doi: 10.3934/cpaa.2012.11.547

[7]

Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407

[8]

Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556

[9]

Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613

[10]

Kazuhiro Ishige, Michinori Ishiwata. Global solutions for a semilinear heat equation in the exterior domain of a compact set. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 847-865. doi: 10.3934/dcds.2012.32.847

[11]

Larissa V. Fardigola. Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control. Mathematical Control & Related Fields, 2013, 3 (2) : 161-183. doi: 10.3934/mcrf.2013.3.161

[12]

Aníbal Rodríguez-Bernal, Alejandro Vidal-López. A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (2) : 635-644. doi: 10.3934/cpaa.2014.13.635

[13]

Jason Murphy, Fabio Pusateri. Almost global existence for cubic nonlinear Schrödinger equations in one space dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2077-2102. doi: 10.3934/dcds.2017089

[14]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141

[15]

Hiroshi Takeda. Global existence of solutions for higher order nonlinear damped wave equations. Conference Publications, 2011, 2011 (Special) : 1358-1367. doi: 10.3934/proc.2011.2011.1358

[16]

Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495

[17]

Hideo Kubo, Kotaro Tsugawa. Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 471-482. doi: 10.3934/dcds.2003.9.471

[18]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273

[19]

Denis Pennequin. Existence of almost periodic solutions of discrete time equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 51-60. doi: 10.3934/dcds.2001.7.51

[20]

Lassaad Aloui, Moez Khenissi. Boundary stabilization of the wave and Schrödinger equations in exterior domains. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 919-934. doi: 10.3934/dcds.2010.27.919

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]