# American Institute of Mathematical Sciences

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. [2] Y. Du, J. Metcalfe, C. 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. [3] V. Georgiev, H. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math., 119 (1997), 1291-1319. [4] R. T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340. [5] R. T. Glassey, Existence in the large for □u = F (u) in two space dimensions, Math. Z., 178 (1981), 233-261. [6] K. Hidano, J. Metcalfe, H. F. Smith, C. D. Sogge and Y. Zhou, On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles, Trans. Amer. Math. Soc., 362 (2010), 2789-2809. [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. [8] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268. [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. [10] A. N. Lai and Y. Zhou, An elementary proof of Strauss conjecture, J. Funct. Anal., 267 (2014), 1364-1381. [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. [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. [13] T. T. Li and Y. Zhou, Nonlinear Wave Equations, Series in Contemporary Mathematics, Shanghai Scientific and Technological Education Publishing House(SSTEPH), 2016. [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. [15] H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math., 118 (1996), 1047-1135. [16] H. Lindblad, J. Metcalfe, C. D. Sogge, M. Tohaneanu and B. C. Wang, The Strauss conjecture on Kerr black hole backgrounds, Math. Ann., 359 (2014), 637-661. [17] M. A. Rammaha, Finite-time blow up for nonlinear wave equations in high dimensions, Comm. Partial Differential Equations, 12 (1987), 677-700. [18] J. Schaeffer, The equation □u = |u|p for the critical value of p, Proc. Roy. Soc. Edinburgh Sect. -A, 101 (1985), 31-44. [19] T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406. [20] H. F. Smith, C. 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. [21] C. D. Sogge and B. C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math., 112 (2010), 1-32. [22] W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133. [23] H. Takamura, An elementary proof of the exponential blow-up for semi-linear wave equations, Math. Methods Appl. Sci., 17 (1994), 239-249. [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. [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. [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. [27] B. C. Wang and X. Yu, Concerning the Strauss conjecture on asymptotically Euclidean manifolds, J. Math. Anal. Appl., 379 (2011), 549-566. [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. [29] X. Yu, Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture, Differential and Integral Equations, 24 (2011), 443-468. [30] Y. Zhou, Cauchy problem for semilinear wave equations with small data in four space dimensions, J. Partial Differential Equations, 8 (1995), 135-144. [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. [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.

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. [2] Y. Du, J. Metcalfe, C. 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. [3] V. Georgiev, H. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math., 119 (1997), 1291-1319. [4] R. T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340. [5] R. T. Glassey, Existence in the large for □u = F (u) in two space dimensions, Math. Z., 178 (1981), 233-261. [6] K. Hidano, J. Metcalfe, H. F. Smith, C. D. Sogge and Y. Zhou, On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles, Trans. Amer. Math. Soc., 362 (2010), 2789-2809. [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. [8] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268. [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. [10] A. N. Lai and Y. Zhou, An elementary proof of Strauss conjecture, J. Funct. Anal., 267 (2014), 1364-1381. [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. [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. [13] T. T. Li and Y. Zhou, Nonlinear Wave Equations, Series in Contemporary Mathematics, Shanghai Scientific and Technological Education Publishing House(SSTEPH), 2016. [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. [15] H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math., 118 (1996), 1047-1135. [16] H. Lindblad, J. Metcalfe, C. D. Sogge, M. Tohaneanu and B. C. Wang, The Strauss conjecture on Kerr black hole backgrounds, Math. Ann., 359 (2014), 637-661. [17] M. A. Rammaha, Finite-time blow up for nonlinear wave equations in high dimensions, Comm. Partial Differential Equations, 12 (1987), 677-700. [18] J. Schaeffer, The equation □u = |u|p for the critical value of p, Proc. Roy. Soc. Edinburgh Sect. -A, 101 (1985), 31-44. [19] T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406. [20] H. F. Smith, C. 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. [21] C. D. Sogge and B. C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math., 112 (2010), 1-32. [22] W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133. [23] H. Takamura, An elementary proof of the exponential blow-up for semi-linear wave equations, Math. Methods Appl. Sci., 17 (1994), 239-249. [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. [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. [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. [27] B. C. Wang and X. Yu, Concerning the Strauss conjecture on asymptotically Euclidean manifolds, J. Math. Anal. Appl., 379 (2011), 549-566. [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. [29] X. Yu, Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture, Differential and Integral Equations, 24 (2011), 443-468. [30] Y. Zhou, Cauchy problem for semilinear wave equations with small data in four space dimensions, J. Partial Differential Equations, 8 (1995), 135-144. [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. [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.
 [1] Kyouhei Wakasa. Blow-up of solutions to semilinear wave equations with non-zero initial data. Conference Publications, 2015, 2015 (special) : 1105-1114. doi: 10.3934/proc.2015.1105 [2] Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure & Applied Analysis, 2011, 10 (1) : 161-177. doi: 10.3934/cpaa.2011.10.161 [3] Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103 [4] Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315 [5] Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101 [6] Hiroyuki Takamura, Hiroshi Uesaka, Kyouhei Wakasa. Sharp blow-up for semilinear wave equations with non-compactly supported data. Conference Publications, 2011, 2011 (Special) : 1351-1357. doi: 10.3934/proc.2011.2011.1351 [7] Júlia Matos. Unfocused blow up solutions of semilinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 905-928. doi: 10.3934/dcds.1999.5.905 [8] Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431 [9] Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n}$. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319 [10] Can Gao, Joachim Krieger. Optimal polynomial blow up range for critical wave maps. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1705-1741. doi: 10.3934/cpaa.2015.14.1705 [11] Dongfen Bian. Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1591-1611. doi: 10.3934/dcdss.2016065 [12] Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135 [13] Yacheng Liu, Runzhang Xu. Potential well method for initial boundary value problem of the generalized double dispersion equations. Communications on Pure & Applied Analysis, 2008, 7 (1) : 63-81. doi: 10.3934/cpaa.2008.7.63 [14] Jitao Liu. On the initial boundary value problem for certain 2D MHD-$\alpha$ equations without velocity viscosity. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1179-1191. doi: 10.3934/cpaa.2016.15.1179 [15] Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399 [16] Yohei Fujishima. On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 449-475. doi: 10.3934/cpaa.2018025 [17] Van Tien Nguyen. On the blow-up results for a class of strongly perturbed semilinear heat equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3585-3626. doi: 10.3934/dcds.2015.35.3585 [18] Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733 [19] Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121 [20] Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212

2018 Impact Factor: 0.925