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September  2020, 40(9): 5513-5540. doi: 10.3934/dcds.2020236

Nonexistence of global solutions for the semilinear Moore – Gibson – Thompson equation in the conservative case

1. 

Institute of Applied Analysis, Faculty of Mathematics and Computer Science, Technical University Bergakademie Freiberg, Prüferstraße 9, 09596 Freiberg, Germany

2. 

Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy

* Corresponding author: Alessandro Palmieri

Received  December 2019 Revised  March 2020 Published  June 2020

Fund Project: The Ph.D. study of the first author is supported by Sächsiches Landesgraduiertenstipendium. The second author is supported by the University of Pisa, Project PRA 2018 49

In this work, the Cauchy problem for the semilinear Moore – Gibson – Thompson (MGT) equation with power nonlinearity $ |u|^p $ on the right – hand side is studied. Applying $ L^2 - L^2 $ estimates and a fixed point theorem, we obtain local (in time) existence of solutions to the semilinear MGT equation. Then, the blow - up of local in time solutions is proved by using an iteration method, under certain sign assumption for initial data, and providing that the exponent of the power of the nonlinearity fulfills $ 1<p\leqslant p_{\mathrm{Str}}(n) $ for $ n\geqslant2 $ and $ p>1 $ for $ n = 1 $. Here the Strauss exponent $ p_{\mathrm{Str}}(n) $ is the critical exponent for the semilinear wave equation with power nonlinearity. In particular, in the limit case $ p = p_{\mathrm{Str}}(n) $ a different approach with a weighted space average of a local in time solution is considered.

Citation: Wenhui Chen, Alessandro Palmieri. Nonexistence of global solutions for the semilinear Moore – Gibson – Thompson equation in the conservative case. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5513-5540. doi: 10.3934/dcds.2020236
References:
[1]

R. AgemiY. Kurokawa and H. Takamura, Critical curve for $p$-$q$ systems of nonlinear wave equations in three space dimensions, J. Differential Equations, 167 (2000), 87-133.  doi: 10.1006/jdeq.2000.3766.

[2]

M. O. Alves, A. H. Caixeta, M. A. J. Silva and J. H. Rodrigues, Moore-Gibson-Thompson equation with memory in a history framework: A semigroup approach, Z. Angew. Math. Phys., 69 (2018), 19pp. doi: 10.1007/s00033-018-0999-5.

[3]

A. H. CaixetaI. Lasiecka and V. N. Domingos Cavalcanti, On long time behavior of Moore-Gibson-Thompson equation with molecular relaxation, Evol. Equ. Control Theory, 5 (2016), 661-676.  doi: 10.3934/eect.2016024.

[4]

W. Chen and A. Palmieri, A blow-up result for the semilinear Moore-Gibson-Thompson equation with nonlinearity of derivative type in the conservative case, preprint, arXiv: 1909.09348.

[5]

F. Dell'Oro, I. Lasiecka and V. Pata, On the MGT equation with memory of type Ⅱ, preprint, arXiv: 1904.08203.

[6]

F. Dell'OroI. Lasiecka and V. Pata, The Moore-Gibson-Thompson equation with memory in the critical case, J. Differential Equations, 261 (2016), 4188-4222.  doi: 10.1016/j.jde.2016.06.025.

[7]

F. Dell'Oro and V. Pata, On the Moore-Gibson-Thompson equation and its relation to linear viscoelasticity, Appl. Math. Optim., 76 (2017), 641-655.  doi: 10.1007/s00245-016-9365-1.

[8]

P. M. N. DharmawardaneJ. E. Muñoz Rivera and S. Kawashima, Decay property for second order hyperbolic systems of viscoelastic materials, J. Math. Anal. Appl., 366 (2010), 621-635.  doi: 10.1016/j.jmaa.2009.12.019.

[9]

S. Di Pomponio and V. Georgiev, Life-span of subcritical semilinear wave equation, Asymptot. Anal., 28 (2001), 91-114. 

[10]

M. R. Ebert and M. Reissig, Methods for Partial Differential Equations. Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models, Birkhäuser/Springer, Cham, 2018. doi: 10.1007/978-3-319-66456-9.

[11]

V. GeorgievH. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math., 119 (1997), 1291-1319.  doi: 10.1353/ajm.1997.0038.

[12]

R. T. Glassey, Existence in the large for $\square u = F(u)$ in two space dimensions, Math. Z., 178 (1981), 233-261.  doi: 10.1007/BF01262042.

[13]

R. T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.  doi: 10.1007/BF01162066.

[14]

G. C. Gorain, Stabilization for the vibrations modeled by the 'standard linear model' of viscoelasticity, Proc. Indian Acad. Sci. Math. Sci., 120 (2010), 495-506.  doi: 10.1007/s12044-010-0038-8.

[15]

T. Imai, M. Kato, H. Takamura and K. Wakasa, The sharp lower bound of the lifespan of solutions to semilinear wave equations with low powers in two space dimensions, in Adv. Stud. Pure Math., Asymptotic Analysis for Nonlinear Dispersive and Wave Equations, Mathematical Society of Japan, 2019, 31–53. doi: 10.2969/aspm/08110031.

[16]

H. Jiao and Z. Zhou, An elementary proof of the blow-up for semilinear wave equation in high space dimensions, J. Differential Equations, 189 (2003), 355-365.  doi: 10.1016/S0022-0396(02)00041-4.

[17]

F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.  doi: 10.1007/BF01647974.

[18]

P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189.

[19]

B. Kaltenbacher and I. Lasiecka, Exponential decay for low and higher energies in the third order linear Moore-Gibson-Thompson equation with variable viscosity, Palest. J. Math., 1 (2012), 1-10. 

[20]

B. KaltenbacherI. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control Cybernet., 40 (2011), 971-988. 

[21]

T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math., 33 (1980), 501-505.  doi: 10.1002/cpa.3160330403.

[22]

N.-A. Lai and Y. Zhou, An elementary proof of Strauss conjecture, J. Funct. Anal., 267 (2014), 1364-1381.  doi: 10.1016/j.jfa.2014.05.020.

[23]

I. Lasiecka, Global solvability of Moore-Gibson-Thompson equation with memory arising in nonlinear acoustics, J. Evol. Equ., 17 (2017), 411-441.  doi: 10.1007/s00028-016-0353-3.

[24]

I. Lasiecka and X. Wang, Moore-Gibson-Thompson equation with memory, part Ⅰ: Exponential decay of energy, Z. Angew. Math. Phys., 67 (2016), 23pp. doi: 10.1007/s00033-015-0597-8.

[25]

I. Lasiecka and X. Wang, Moore-Gibson-Thompson equation with memory, part Ⅱ: General decay of energy, J. Differential Equations, 259 (2015), 7610-7635.  doi: 10.1016/j.jde.2015.08.052.

[26]

H. Lindblad, Blow-up for solutions of $\square u = |u|^p$ with small initial data, Comm. Partial Differential Equations, 15 (1990), 757-821.  doi: 10.1080/03605309908820708.

[27]

H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math., 118 (1996), 1047-1135.  doi: 10.1353/ajm.1996.0042.

[28]

R. MarchandT. McDevitt and R. Triggiani, An abstract semigroup approach to the third-order Moore-Gibson-Thompson partial differential equation arising in high-intensity ultrasound: Structural decomposition, spectral analysis, exponential stability, Math. Methods Appl. Sci., 35 (2012), 1896-1929.  doi: 10.1002/mma.1576.

[29]

F. K. Moore and W. E. Gibson, Propagation of weak disturbances in a gas subject to relaxation effects, J. Aero/Space Sci., 27 (1960), 117-127.  doi: 10.2514/8.8418.

[30]

A. Palmieri and H. Takamura, Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities, Nonlinear Anal., 187 (2019), 467-492.  doi: 10.1016/j.na.2019.06.016.

[31]

A. Palmieri and H. Takamura, Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations of derivative type in the scattering case, Mediterr. J. Math., 17 (2020), 20pp. doi: 10.1007/s00009-019-1445-4.

[32]

A. Palmieri and H. Takamura, Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations in the scattering case with mixed nonlinear terms, preprint, arXiv: 1901.04038.

[33]

A. Palmieri and Z. Tu, A blow-up result for a semilinear wave equation with scale-invariant damping and mass and nonlinearity of derivative type, preprint, arXiv: 1905.11025v2.

[34]

M. Pellicer and B. Said-Houari, Wellposedness and decay rates for the Cauchy problem of the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Appl. Math. Optim., 80 (2019), 447-478.  doi: 10.1007/s00245-017-9471-8.

[35]

M. Pellicer and J. Solá-Morales, Optimal scalar products in the Moore-Gibson-Thompson equation, Evol. Equ. Control Theory, 8 (2019), 203-220.  doi: 10.3934/eect.2019011.

[36]

R. Racke and B. Said-Houari, Decay rates for semilinear viscoelastic systems in weighted spaces, J. Hyperbolic Differ. Equ., 9 (2012), 67-103.  doi: 10.1142/S0219891612500026.

[37]

J. Schaeffer, The equation $u_tt-\Delta u = |u|^p$ for the critical value of $p$, Proc. Roy. Soc. Edinburgh Sect. A., 101 (1985), 31-44.  doi: 10.1017/S0308210500026135.

[38]

T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406.  doi: 10.1016/0022-0396(84)90169-4.

[39]

W. A. Strauss, Nonlinear scattering theory at low energy, J. Functional Analysis, 41 (1981), 110-133.  doi: 10.1016/0022-1236(81)90063-X.

[40]

H. Takamura, Improved Kato's lemma on ordinary differential inequality and its application to semilinear wave equations, Nonlinear Anal., 125 (2015), 227-240.  doi: 10.1016/j.na.2015.05.024.

[41]

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.  doi: 10.1016/j.jde.2011.03.024.

[42]

D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc., 353 (2001), 795-807.  doi: 10.1090/S0002-9947-00-02750-1.

[43]

P. A. Thompson, Compressible Fluid Dynamics, McGraw-Hill, New York, 1972.

[44]

K. Wakasa and B. Yordanov, Blow-up of solutions to critical semilinear wave equations with variable coefficients, J. Differential Equations, 266 (2019), 5360-5376.  doi: 10.1016/j.jde.2018.10.028.

[45]

K. Wakasa and B. Yordanov, On the nonexistence of global solutions for critical semilinear wave equations with damping in the scattering case, Nonlinear Anal., 180 (2019), 67-74.  doi: 10.1016/j.na.2018.09.012.

[46]

B. T. Yordanov and Q. S. Zhang, Finite time blow up for critical wave equations in high dimensions, J. Funct. Anal., 231 (2006), 361-374.  doi: 10.1016/j.jfa.2005.03.012.

[47]

Y. Zhou, Life span of classical solutions to $u_tt-u_xx=|u|^{1+\alpha}$, Chinese Ann. Math. Ser. B, 13 (1992), 230-243. 

[48]

Y. Zhou, Blow up of classical solutions to $\square u=|u|^{1+\alpha}$ in three space dimensions, J. Partial Differential Equations, 5 (1992), 21-32. 

[49]

Y. Zhou, Life span of classical solutions to $\square u=|u|^p$ in two space dimensions, Chinese Ann. Math. Ser. B, 14 (1993), 225-236. 

[50]

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

[51]

Y. Zhou, Blow up of solutions to semilinear wave equations with critical exponent in high dimensions, Chin. Ann. Math. Ser. B, 28 (2007), 205-212.  doi: 10.1007/s11401-005-0205-x.

[52]

Y. Zhou and W. Han, Life-span of solutions to critical semilinear wave equations, Comm. Partial Differential Equations, 39 (2014), 439-451.  doi: 10.1080/03605302.2013.863914.

show all references

References:
[1]

R. AgemiY. Kurokawa and H. Takamura, Critical curve for $p$-$q$ systems of nonlinear wave equations in three space dimensions, J. Differential Equations, 167 (2000), 87-133.  doi: 10.1006/jdeq.2000.3766.

[2]

M. O. Alves, A. H. Caixeta, M. A. J. Silva and J. H. Rodrigues, Moore-Gibson-Thompson equation with memory in a history framework: A semigroup approach, Z. Angew. Math. Phys., 69 (2018), 19pp. doi: 10.1007/s00033-018-0999-5.

[3]

A. H. CaixetaI. Lasiecka and V. N. Domingos Cavalcanti, On long time behavior of Moore-Gibson-Thompson equation with molecular relaxation, Evol. Equ. Control Theory, 5 (2016), 661-676.  doi: 10.3934/eect.2016024.

[4]

W. Chen and A. Palmieri, A blow-up result for the semilinear Moore-Gibson-Thompson equation with nonlinearity of derivative type in the conservative case, preprint, arXiv: 1909.09348.

[5]

F. Dell'Oro, I. Lasiecka and V. Pata, On the MGT equation with memory of type Ⅱ, preprint, arXiv: 1904.08203.

[6]

F. Dell'OroI. Lasiecka and V. Pata, The Moore-Gibson-Thompson equation with memory in the critical case, J. Differential Equations, 261 (2016), 4188-4222.  doi: 10.1016/j.jde.2016.06.025.

[7]

F. Dell'Oro and V. Pata, On the Moore-Gibson-Thompson equation and its relation to linear viscoelasticity, Appl. Math. Optim., 76 (2017), 641-655.  doi: 10.1007/s00245-016-9365-1.

[8]

P. M. N. DharmawardaneJ. E. Muñoz Rivera and S. Kawashima, Decay property for second order hyperbolic systems of viscoelastic materials, J. Math. Anal. Appl., 366 (2010), 621-635.  doi: 10.1016/j.jmaa.2009.12.019.

[9]

S. Di Pomponio and V. Georgiev, Life-span of subcritical semilinear wave equation, Asymptot. Anal., 28 (2001), 91-114. 

[10]

M. R. Ebert and M. Reissig, Methods for Partial Differential Equations. Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models, Birkhäuser/Springer, Cham, 2018. doi: 10.1007/978-3-319-66456-9.

[11]

V. GeorgievH. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math., 119 (1997), 1291-1319.  doi: 10.1353/ajm.1997.0038.

[12]

R. T. Glassey, Existence in the large for $\square u = F(u)$ in two space dimensions, Math. Z., 178 (1981), 233-261.  doi: 10.1007/BF01262042.

[13]

R. T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.  doi: 10.1007/BF01162066.

[14]

G. C. Gorain, Stabilization for the vibrations modeled by the 'standard linear model' of viscoelasticity, Proc. Indian Acad. Sci. Math. Sci., 120 (2010), 495-506.  doi: 10.1007/s12044-010-0038-8.

[15]

T. Imai, M. Kato, H. Takamura and K. Wakasa, The sharp lower bound of the lifespan of solutions to semilinear wave equations with low powers in two space dimensions, in Adv. Stud. Pure Math., Asymptotic Analysis for Nonlinear Dispersive and Wave Equations, Mathematical Society of Japan, 2019, 31–53. doi: 10.2969/aspm/08110031.

[16]

H. Jiao and Z. Zhou, An elementary proof of the blow-up for semilinear wave equation in high space dimensions, J. Differential Equations, 189 (2003), 355-365.  doi: 10.1016/S0022-0396(02)00041-4.

[17]

F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.  doi: 10.1007/BF01647974.

[18]

P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189.

[19]

B. Kaltenbacher and I. Lasiecka, Exponential decay for low and higher energies in the third order linear Moore-Gibson-Thompson equation with variable viscosity, Palest. J. Math., 1 (2012), 1-10. 

[20]

B. KaltenbacherI. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control Cybernet., 40 (2011), 971-988. 

[21]

T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math., 33 (1980), 501-505.  doi: 10.1002/cpa.3160330403.

[22]

N.-A. Lai and Y. Zhou, An elementary proof of Strauss conjecture, J. Funct. Anal., 267 (2014), 1364-1381.  doi: 10.1016/j.jfa.2014.05.020.

[23]

I. Lasiecka, Global solvability of Moore-Gibson-Thompson equation with memory arising in nonlinear acoustics, J. Evol. Equ., 17 (2017), 411-441.  doi: 10.1007/s00028-016-0353-3.

[24]

I. Lasiecka and X. Wang, Moore-Gibson-Thompson equation with memory, part Ⅰ: Exponential decay of energy, Z. Angew. Math. Phys., 67 (2016), 23pp. doi: 10.1007/s00033-015-0597-8.

[25]

I. Lasiecka and X. Wang, Moore-Gibson-Thompson equation with memory, part Ⅱ: General decay of energy, J. Differential Equations, 259 (2015), 7610-7635.  doi: 10.1016/j.jde.2015.08.052.

[26]

H. Lindblad, Blow-up for solutions of $\square u = |u|^p$ with small initial data, Comm. Partial Differential Equations, 15 (1990), 757-821.  doi: 10.1080/03605309908820708.

[27]

H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math., 118 (1996), 1047-1135.  doi: 10.1353/ajm.1996.0042.

[28]

R. MarchandT. McDevitt and R. Triggiani, An abstract semigroup approach to the third-order Moore-Gibson-Thompson partial differential equation arising in high-intensity ultrasound: Structural decomposition, spectral analysis, exponential stability, Math. Methods Appl. Sci., 35 (2012), 1896-1929.  doi: 10.1002/mma.1576.

[29]

F. K. Moore and W. E. Gibson, Propagation of weak disturbances in a gas subject to relaxation effects, J. Aero/Space Sci., 27 (1960), 117-127.  doi: 10.2514/8.8418.

[30]

A. Palmieri and H. Takamura, Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities, Nonlinear Anal., 187 (2019), 467-492.  doi: 10.1016/j.na.2019.06.016.

[31]

A. Palmieri and H. Takamura, Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations of derivative type in the scattering case, Mediterr. J. Math., 17 (2020), 20pp. doi: 10.1007/s00009-019-1445-4.

[32]

A. Palmieri and H. Takamura, Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations in the scattering case with mixed nonlinear terms, preprint, arXiv: 1901.04038.

[33]

A. Palmieri and Z. Tu, A blow-up result for a semilinear wave equation with scale-invariant damping and mass and nonlinearity of derivative type, preprint, arXiv: 1905.11025v2.

[34]

M. Pellicer and B. Said-Houari, Wellposedness and decay rates for the Cauchy problem of the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Appl. Math. Optim., 80 (2019), 447-478.  doi: 10.1007/s00245-017-9471-8.

[35]

M. Pellicer and J. Solá-Morales, Optimal scalar products in the Moore-Gibson-Thompson equation, Evol. Equ. Control Theory, 8 (2019), 203-220.  doi: 10.3934/eect.2019011.

[36]

R. Racke and B. Said-Houari, Decay rates for semilinear viscoelastic systems in weighted spaces, J. Hyperbolic Differ. Equ., 9 (2012), 67-103.  doi: 10.1142/S0219891612500026.

[37]

J. Schaeffer, The equation $u_tt-\Delta u = |u|^p$ for the critical value of $p$, Proc. Roy. Soc. Edinburgh Sect. A., 101 (1985), 31-44.  doi: 10.1017/S0308210500026135.

[38]

T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406.  doi: 10.1016/0022-0396(84)90169-4.

[39]

W. A. Strauss, Nonlinear scattering theory at low energy, J. Functional Analysis, 41 (1981), 110-133.  doi: 10.1016/0022-1236(81)90063-X.

[40]

H. Takamura, Improved Kato's lemma on ordinary differential inequality and its application to semilinear wave equations, Nonlinear Anal., 125 (2015), 227-240.  doi: 10.1016/j.na.2015.05.024.

[41]

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.  doi: 10.1016/j.jde.2011.03.024.

[42]

D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc., 353 (2001), 795-807.  doi: 10.1090/S0002-9947-00-02750-1.

[43]

P. A. Thompson, Compressible Fluid Dynamics, McGraw-Hill, New York, 1972.

[44]

K. Wakasa and B. Yordanov, Blow-up of solutions to critical semilinear wave equations with variable coefficients, J. Differential Equations, 266 (2019), 5360-5376.  doi: 10.1016/j.jde.2018.10.028.

[45]

K. Wakasa and B. Yordanov, On the nonexistence of global solutions for critical semilinear wave equations with damping in the scattering case, Nonlinear Anal., 180 (2019), 67-74.  doi: 10.1016/j.na.2018.09.012.

[46]

B. T. Yordanov and Q. S. Zhang, Finite time blow up for critical wave equations in high dimensions, J. Funct. Anal., 231 (2006), 361-374.  doi: 10.1016/j.jfa.2005.03.012.

[47]

Y. Zhou, Life span of classical solutions to $u_tt-u_xx=|u|^{1+\alpha}$, Chinese Ann. Math. Ser. B, 13 (1992), 230-243. 

[48]

Y. Zhou, Blow up of classical solutions to $\square u=|u|^{1+\alpha}$ in three space dimensions, J. Partial Differential Equations, 5 (1992), 21-32. 

[49]

Y. Zhou, Life span of classical solutions to $\square u=|u|^p$ in two space dimensions, Chinese Ann. Math. Ser. B, 14 (1993), 225-236. 

[50]

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

[51]

Y. Zhou, Blow up of solutions to semilinear wave equations with critical exponent in high dimensions, Chin. Ann. Math. Ser. B, 28 (2007), 205-212.  doi: 10.1007/s11401-005-0205-x.

[52]

Y. Zhou and W. Han, Life-span of solutions to critical semilinear wave equations, Comm. Partial Differential Equations, 39 (2014), 439-451.  doi: 10.1080/03605302.2013.863914.

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