July  2018, 17(4): 1499-1510. doi: 10.3934/cpaa.2018072

Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions

1. 

Institute of Nonlinear Analysis and Department of Mathematics, Lishui University, Lishui 323000, China

2. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

* Corresponding author: Ning-An Lai

Received  September 2016 Revised  May 2017 Published  April 2018

Fund Project: The first author is partially supported by Zhejiang Province Science Foundation(LY18A010008), NSFC(11501273, 11726612, 11771359, 11771194), Chinese Postdoctoral Science Foundation(2017M620128), the Scientific Research Foundation of the First-Class Discipline of Zhejiang Province (B)(201601). The second author is supported by Key Laboratory of Mathematics for Nonlinear Sciences(Fudan University), Ministry of Education of China, Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, NSFC (11726611, 11421061), 973 program (2013CB834100) and 111 project.

This paper focuses on the initial boundary value problem of semilinear wave equation in exterior domain in two space dimensions with critical power. Based on the contradiction argument, we prove that the solution will blow up in a finite time. This complements the existence result of supercritical case by Smith, Sogge and Wang [20] and blow up result of subcritical case by Li and Wang [14] in two space dimensions.

Citation: Ning-An Lai, Yi Zhou. Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1499-1510. doi: 10.3934/cpaa.2018072
References:
[1]

D. Catania and V. Georgiev, Blow-up for the semilinear wave equation in the Schwarzschild metric, Differential Integral Equations, 19 (2006), 799-830.   Google Scholar

[2]

Y. DuJ. MetcalfeC. D. Sogge and Y. Zhou, Concerning the Strauss conjecture and almost global existence for nonlinear Dirichlet-wave equations in 4-dimensions, Comm. Partial Differential Equations, 33 (2008), 1487-1506.   Google Scholar

[3]

V. GeorgievH. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math., 119 (1997), 1291-1319.   Google Scholar

[4]

R. T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.   Google Scholar

[5]

R. T. Glassey, Existence in the large for □u = F (u) in two space dimensions, Math. Z., 178 (1981), 233-261.   Google Scholar

[6]

K. HidanoJ. MetcalfeH. F. SmithC. D. Sogge and Y. Zhou, On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles, Trans. Amer. Math. Soc., 362 (2010), 2789-2809.   Google Scholar

[7]

L. H. Jiao and F. Z. Zhou, An elementary proof of the blow up for semilinear wave equation in high space dimensions, J. Differential Equations, 189 (2003), 355-365.   Google Scholar

[8]

F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.   Google Scholar

[9]

H. Kubo, On the critical decay and power for semilinear wave equations in odd space dimensions, Discrete Contin. Dynam. Systems, 2 (1996), 173-190.   Google Scholar

[10]

A. N. Lai and Y. Zhou, An elementary proof of Strauss conjecture, J. Funct. Anal., 267 (2014), 1364-1381.   Google Scholar

[11]

A. N. Lai and Y. Zhou, Finite time blow up to critical semilinear wave equation outside the ball in 3-D, Nonlinear Anal., 125 (2015), 550-560.   Google Scholar

[12]

A. N. Lai and Y. Zhou, Nonexistence of global solutions to critical semilinear wave equations in exterior domain in high dimensions, Nonlinear Anal., 143 (2016), 89-104.   Google Scholar

[13]

T. T. Li and Y. Zhou, Nonlinear Wave Equations, Series in Contemporary Mathematics, Shanghai Scientific and Technological Education Publishing House(SSTEPH), 2016. Google Scholar

[14]

F. X. Li and X. G. Wang, Blow up of solutions to nonlinear wave equations in 2D exterior domains, Arch. Math., 98 (2012), 265-275.   Google Scholar

[15]

H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math., 118 (1996), 1047-1135.   Google Scholar

[16]

H. LindbladJ. MetcalfeC. D. SoggeM. Tohaneanu and B. C. Wang, The Strauss conjecture on Kerr black hole backgrounds, Math. Ann., 359 (2014), 637-661.   Google Scholar

[17]

M. A. Rammaha, Finite-time blow up for nonlinear wave equations in high dimensions, Comm. Partial Differential Equations, 12 (1987), 677-700.   Google Scholar

[18]

J. Schaeffer, The equation □u = |u|p for the critical value of p, Proc. Roy. Soc. Edinburgh Sect. -A, 101 (1985), 31-44.   Google Scholar

[19]

T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406.   Google Scholar

[20]

H. F. SmithC. D. Sogge and B. C. Wang, Strichartz estimates for Dirichlet-wave equations in two dimensions with applications, Trans. Amer. Math. Soc., 364 (2012), 3329-3347.   Google Scholar

[21]

C. D. Sogge and B. C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math., 112 (2010), 1-32.   Google Scholar

[22]

W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133.   Google Scholar

[23]

H. Takamura, An elementary proof of the exponential blow-up for semi-linear wave equations, Math. Methods Appl. Sci., 17 (1994), 239-249.   Google Scholar

[24]

H. Takamura and K. Wakasa, The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions, J. Differential Equations, 251 (2011), 1157-1171.   Google Scholar

[25]

D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc., 353 (2001), 795-807.   Google Scholar

[26]

B. C. Wang and X. Yu, Recent works on the Strauss conjecture, Recent advances in harmonic analysis and partial differential equations, Contemp. Math., 581, Amer. Math. Soc., Providence, RI, (2012), 235-256. Google Scholar

[27]

B. C. Wang and X. Yu, Concerning the Strauss conjecture on asymptotically Euclidean manifolds, J. Math. Anal. Appl., 379 (2011), 549-566.   Google Scholar

[28]

B. Yordanov and Q. S. Zhang, Finite time blow up for critical wave equations in high dimensions, J. Funct. Anal., 231 (2006), 361-374.   Google Scholar

[29]

X. Yu, Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture, Differential and Integral Equations, 24 (2011), 443-468.   Google Scholar

[30]

Y. Zhou, Cauchy problem for semilinear wave equations with small data in four space dimensions, J. Partial Differential Equations, 8 (1995), 135-144.   Google Scholar

[31]

Y. Zhou, Blow up of solutions to semilinear wave equations with critical exponent in high dimensions, Chinese Ann. Math. Ser. -B, 28 (2007), 205-212.   Google Scholar

[32]

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.   Google Scholar

show all references

References:
[1]

D. Catania and V. Georgiev, Blow-up for the semilinear wave equation in the Schwarzschild metric, Differential Integral Equations, 19 (2006), 799-830.   Google Scholar

[2]

Y. DuJ. MetcalfeC. D. Sogge and Y. Zhou, Concerning the Strauss conjecture and almost global existence for nonlinear Dirichlet-wave equations in 4-dimensions, Comm. Partial Differential Equations, 33 (2008), 1487-1506.   Google Scholar

[3]

V. GeorgievH. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math., 119 (1997), 1291-1319.   Google Scholar

[4]

R. T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.   Google Scholar

[5]

R. T. Glassey, Existence in the large for □u = F (u) in two space dimensions, Math. Z., 178 (1981), 233-261.   Google Scholar

[6]

K. HidanoJ. MetcalfeH. F. SmithC. D. Sogge and Y. Zhou, On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles, Trans. Amer. Math. Soc., 362 (2010), 2789-2809.   Google Scholar

[7]

L. H. Jiao and F. Z. Zhou, An elementary proof of the blow up for semilinear wave equation in high space dimensions, J. Differential Equations, 189 (2003), 355-365.   Google Scholar

[8]

F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.   Google Scholar

[9]

H. Kubo, On the critical decay and power for semilinear wave equations in odd space dimensions, Discrete Contin. Dynam. Systems, 2 (1996), 173-190.   Google Scholar

[10]

A. N. Lai and Y. Zhou, An elementary proof of Strauss conjecture, J. Funct. Anal., 267 (2014), 1364-1381.   Google Scholar

[11]

A. N. Lai and Y. Zhou, Finite time blow up to critical semilinear wave equation outside the ball in 3-D, Nonlinear Anal., 125 (2015), 550-560.   Google Scholar

[12]

A. N. Lai and Y. Zhou, Nonexistence of global solutions to critical semilinear wave equations in exterior domain in high dimensions, Nonlinear Anal., 143 (2016), 89-104.   Google Scholar

[13]

T. T. Li and Y. Zhou, Nonlinear Wave Equations, Series in Contemporary Mathematics, Shanghai Scientific and Technological Education Publishing House(SSTEPH), 2016. Google Scholar

[14]

F. X. Li and X. G. Wang, Blow up of solutions to nonlinear wave equations in 2D exterior domains, Arch. Math., 98 (2012), 265-275.   Google Scholar

[15]

H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math., 118 (1996), 1047-1135.   Google Scholar

[16]

H. LindbladJ. MetcalfeC. D. SoggeM. Tohaneanu and B. C. Wang, The Strauss conjecture on Kerr black hole backgrounds, Math. Ann., 359 (2014), 637-661.   Google Scholar

[17]

M. A. Rammaha, Finite-time blow up for nonlinear wave equations in high dimensions, Comm. Partial Differential Equations, 12 (1987), 677-700.   Google Scholar

[18]

J. Schaeffer, The equation □u = |u|p for the critical value of p, Proc. Roy. Soc. Edinburgh Sect. -A, 101 (1985), 31-44.   Google Scholar

[19]

T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406.   Google Scholar

[20]

H. F. SmithC. D. Sogge and B. C. Wang, Strichartz estimates for Dirichlet-wave equations in two dimensions with applications, Trans. Amer. Math. Soc., 364 (2012), 3329-3347.   Google Scholar

[21]

C. D. Sogge and B. C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math., 112 (2010), 1-32.   Google Scholar

[22]

W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133.   Google Scholar

[23]

H. Takamura, An elementary proof of the exponential blow-up for semi-linear wave equations, Math. Methods Appl. Sci., 17 (1994), 239-249.   Google Scholar

[24]

H. Takamura and K. Wakasa, The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions, J. Differential Equations, 251 (2011), 1157-1171.   Google Scholar

[25]

D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc., 353 (2001), 795-807.   Google Scholar

[26]

B. C. Wang and X. Yu, Recent works on the Strauss conjecture, Recent advances in harmonic analysis and partial differential equations, Contemp. Math., 581, Amer. Math. Soc., Providence, RI, (2012), 235-256. Google Scholar

[27]

B. C. Wang and X. Yu, Concerning the Strauss conjecture on asymptotically Euclidean manifolds, J. Math. Anal. Appl., 379 (2011), 549-566.   Google Scholar

[28]

B. Yordanov and Q. S. Zhang, Finite time blow up for critical wave equations in high dimensions, J. Funct. Anal., 231 (2006), 361-374.   Google Scholar

[29]

X. Yu, Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture, Differential and Integral Equations, 24 (2011), 443-468.   Google Scholar

[30]

Y. Zhou, Cauchy problem for semilinear wave equations with small data in four space dimensions, J. Partial Differential Equations, 8 (1995), 135-144.   Google Scholar

[31]

Y. Zhou, Blow up of solutions to semilinear wave equations with critical exponent in high dimensions, Chinese Ann. Math. Ser. -B, 28 (2007), 205-212.   Google Scholar

[32]

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.   Google Scholar

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