doi: 10.3934/eect.2020085

A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case

1. 

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

2. 

Department of Mathematics, University of Pisa, 56127, Italy

* Corresponding author: Wenhui Chen

Received  January 2020 Revised  June 2020 Published  August 2020

In this paper, we study the blow – up of solutions to the semilinear Moore – Gibson – Thompson (MGT) equation with nonlinearity of derivative type $ |u_t|^p $ in the conservative case. We apply an iteration method in order to study both the subcritical case and the critical case. Hence, we obtain a blow – up result for the semilinear MGT equation (under suitable assumptions for initial data) when the exponent $ p $ for the nonlinear term satisfies $ 1<p\leqslant (n+1)/(n-1) $ for $ n\geqslant2 $ and $ p>1 $ for $ n = 1 $. In particular, we find the same blow – up range for $ p $ as in the corresponding semilinear wave equation with nonlinearity of derivative type.

Citation: Wenhui Chen, Alessandro Palmieri. A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case. Evolution Equations & Control Theory, doi: 10.3934/eect.2020085
References:
[1]

R. Agemi, Blow-up of solutions to nonlinear wave equations in two space dimensions, Manuscripta Math., 73 (1991), 153-162.  doi: 10.1007/BF02567635.  Google Scholar

[2]

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.  Google Scholar

[3]

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), 19. doi: 10.1007/s00033-018-0999-5.  Google Scholar

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F. Bucci and M. Eller, The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation, preprint, (2020), arXiv: 2004.11167. Google Scholar

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F. Bucci and I. Lasiecka, Feedback control of the acoustic pressure in ultrasonic wave propagation, Optimization, 68 (2019), 1811-1854.  doi: 10.1080/02331934.2018.1504051.  Google Scholar

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F. Bucci and L. Pandolfi, On the regularity of solutions to the Moore-Gibson-Thompson equation: A perspective via wave equations with memory, J. Evol. Equ., (2019). doi: 10.1007/s00028-019-00549-x.  Google Scholar

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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.  Google Scholar

[8]

W. Chen and R. Ikehata, The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case, preprint, (2020), arXiv: 2006.00758v2. Google Scholar

[9]

W. Chen and A. Palmieri, Nonexistence of global solutions for the semilinear Moore – Gibson – Thompson equation in the conservative case, Discrete Contin. Dyn. Syst., 40 (2020), 5513-5540.  doi: 10.3934/dcds.2020236.  Google Scholar

[10]

F. Dell'Oro, I. Lasiecka and V. Pata, A note on the Moore-Gibson-Thompson equation with memory of type Ⅱ, J. Evol. Equ., (2019). doi: 10.1007/s00028-019-00554-0.  Google Scholar

[11]

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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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K. Hidano and K. Tsutaya, Global existence and asymptotic behavior of solutions for nonlinear wave equations, Indiana Univ. Math. J., 44 (1995), 1273-1305.  doi: 10.1512/iumj.1995.44.2028.  Google Scholar

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K. HidanoC. Wang and K. Yokoyama, The Glassey conjecture with radially symmetric data, J. Math. Pures Appl., 98 (2012), 518-541.  doi: 10.1016/j.matpur.2012.01.007.  Google Scholar

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M. Ikeda, Z. Tu and K. Wakasa, Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass, preprint, (2019), arXiv: 1904.09574. Google Scholar

[17]

F. John, Blow-up for quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51.  doi: 10.1002/cpa.3160340103.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

[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.   Google Scholar

[21]

N.-A. Lai and H. Takamura, Blow-up for semilinear damped wave equations with subcritical exponent in the scattering case, Nonlinear Anal., 168 (2018), 222-237.  doi: 10.1016/j.na.2017.12.008.  Google Scholar

[22]

N.-A. Lai and H. Takamura, Nonexistence of global solutions of nonlinear wave equations with weak time-dependent damping related to Glassey's conjecture, Differential Integral Equations, 32 (2019), 37–48. https://projecteuclid.org/euclid.die/1544497285.  Google Scholar

[23]

N.-A. Lai and H. Takamura, Nonexistence of global solutions of wave equations with weak time-dependent damping and combined nonlinearity, Nonlinear Anal. Real World Appl., 45 (2019), 83-96.  doi: 10.1016/j.nonrwa.2018.06.008.  Google Scholar

[24]

N.-A. LaiH. Takamura and K. Wakasa, Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent, J. Differential Equations, 263 (2017), 5377-5394.  doi: 10.1016/j.jde.2017.06.017.  Google Scholar

[25]

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.  Google Scholar

[26]

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

[27]

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.  Google Scholar

[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.  Google Scholar

[29]

K. Masuda, Blow-up solutions for quasilinear wave equations in two space dimensions, North-Holland Math. Stud., 98 (1984), 87-91.  doi: 10.1016/S0304-0208(08)71493-2.  Google Scholar

[30]

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

[31]

A. Palmieri, A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale-invariant lower order terms, Math. Methods Appl. Sci., 43 (2020). doi: 10.1002/mma.6412.  Google Scholar

[32]

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.  Google Scholar

[33]

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), 13, 20 pp. doi: 10.1007/s00009-019-1445-4.  Google Scholar

[34]

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. Google Scholar

[35]

A. Palmieri and Z. Tu, Lifespan of semilinear wave equation with scale invariant dissipation and mass and sub-Strauss power nonlinearity, J. Math. Anal. Appl., 470 (2019), 447-469.  doi: 10.1016/j.jmaa.2018.10.015.  Google Scholar

[36]

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. Google Scholar

[37]

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.  Google Scholar

[38]

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

[39]

R. Racke and B. Said-Houari, Global well-posedness of the Cauchy problem for the Jordan-Moore-Gibson-Thompson equation, preprint, http://nbn-resolving.de/urn:nbn:de:bsz:352-2-8ztzhsco3jj82 Google Scholar

[40]

M. A. Rammaha, Finite-time blow-up for nonlinear wave equations in high dimensions, Comm. Partial Differential Equations, 12 (1987), 677-700.  doi: 10.1080/03605308708820506.  Google Scholar

[41]

J. Schaeffer, Finite-time blow-up for $u_tt-\Delta u = H(u_r, u_t)$, Comm. Partial Differential Equations, 11 (1986), 513-543.  doi: 10.1080/03605308608820434.  Google Scholar

[42]

T. C. Sideris, Global behavior of solutions to nonlinear wave equations in three dimensions, Comm. Partial Differential Equations, 8 (1983), 1291-1323.  doi: 10.1080/03605308308820304.  Google Scholar

[43]

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.  Google Scholar

[44]

H. Takamura and K. Wakasa, Almost global solutions of semilinear wave equations with the critical exponent in high dimensions, Nonlinear Anal., 109 (2014), 187-229.  doi: 10.1016/j.na.2014.06.007.  Google Scholar

[45]

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

[46]

N. Tzvetkov, Existence of global solutions to nonlinear massless Dirac system and wave equation with small data, Tsukuba J. Math., 22 (1998), 193-211.  doi: 10.21099/tkbjm/1496163480.  Google Scholar

[47]

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.  Google Scholar

[48]

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.  Google Scholar

[49]

Y. Zhou, Blow up of solutions to the Cauchy problem for nonlinear wave equations, Chinese Ann. Math. Ser. B, 22 (2001), 275-280.  doi: 10.1142/S0252959901000280.  Google Scholar

show all references

References:
[1]

R. Agemi, Blow-up of solutions to nonlinear wave equations in two space dimensions, Manuscripta Math., 73 (1991), 153-162.  doi: 10.1007/BF02567635.  Google Scholar

[2]

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.  Google Scholar

[3]

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), 19. doi: 10.1007/s00033-018-0999-5.  Google Scholar

[4]

F. Bucci and M. Eller, The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation, preprint, (2020), arXiv: 2004.11167. Google Scholar

[5]

F. Bucci and I. Lasiecka, Feedback control of the acoustic pressure in ultrasonic wave propagation, Optimization, 68 (2019), 1811-1854.  doi: 10.1080/02331934.2018.1504051.  Google Scholar

[6]

F. Bucci and L. Pandolfi, On the regularity of solutions to the Moore-Gibson-Thompson equation: A perspective via wave equations with memory, J. Evol. Equ., (2019). doi: 10.1007/s00028-019-00549-x.  Google Scholar

[7]

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.  Google Scholar

[8]

W. Chen and R. Ikehata, The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case, preprint, (2020), arXiv: 2006.00758v2. Google Scholar

[9]

W. Chen and A. Palmieri, Nonexistence of global solutions for the semilinear Moore – Gibson – Thompson equation in the conservative case, Discrete Contin. Dyn. Syst., 40 (2020), 5513-5540.  doi: 10.3934/dcds.2020236.  Google Scholar

[10]

F. Dell'Oro, I. Lasiecka and V. Pata, A note on the Moore-Gibson-Thompson equation with memory of type Ⅱ, J. Evol. Equ., (2019). doi: 10.1007/s00028-019-00554-0.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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.  Google Scholar

[14]

K. Hidano and K. Tsutaya, Global existence and asymptotic behavior of solutions for nonlinear wave equations, Indiana Univ. Math. J., 44 (1995), 1273-1305.  doi: 10.1512/iumj.1995.44.2028.  Google Scholar

[15]

K. HidanoC. Wang and K. Yokoyama, The Glassey conjecture with radially symmetric data, J. Math. Pures Appl., 98 (2012), 518-541.  doi: 10.1016/j.matpur.2012.01.007.  Google Scholar

[16]

M. Ikeda, Z. Tu and K. Wakasa, Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass, preprint, (2019), arXiv: 1904.09574. Google Scholar

[17]

F. John, Blow-up for quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51.  doi: 10.1002/cpa.3160340103.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[21]

N.-A. Lai and H. Takamura, Blow-up for semilinear damped wave equations with subcritical exponent in the scattering case, Nonlinear Anal., 168 (2018), 222-237.  doi: 10.1016/j.na.2017.12.008.  Google Scholar

[22]

N.-A. Lai and H. Takamura, Nonexistence of global solutions of nonlinear wave equations with weak time-dependent damping related to Glassey's conjecture, Differential Integral Equations, 32 (2019), 37–48. https://projecteuclid.org/euclid.die/1544497285.  Google Scholar

[23]

N.-A. Lai and H. Takamura, Nonexistence of global solutions of wave equations with weak time-dependent damping and combined nonlinearity, Nonlinear Anal. Real World Appl., 45 (2019), 83-96.  doi: 10.1016/j.nonrwa.2018.06.008.  Google Scholar

[24]

N.-A. LaiH. Takamura and K. Wakasa, Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent, J. Differential Equations, 263 (2017), 5377-5394.  doi: 10.1016/j.jde.2017.06.017.  Google Scholar

[25]

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.  Google Scholar

[26]

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

[27]

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.  Google Scholar

[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.  Google Scholar

[29]

K. Masuda, Blow-up solutions for quasilinear wave equations in two space dimensions, North-Holland Math. Stud., 98 (1984), 87-91.  doi: 10.1016/S0304-0208(08)71493-2.  Google Scholar

[30]

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

[31]

A. Palmieri, A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale-invariant lower order terms, Math. Methods Appl. Sci., 43 (2020). doi: 10.1002/mma.6412.  Google Scholar

[32]

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.  Google Scholar

[33]

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), 13, 20 pp. doi: 10.1007/s00009-019-1445-4.  Google Scholar

[34]

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. Google Scholar

[35]

A. Palmieri and Z. Tu, Lifespan of semilinear wave equation with scale invariant dissipation and mass and sub-Strauss power nonlinearity, J. Math. Anal. Appl., 470 (2019), 447-469.  doi: 10.1016/j.jmaa.2018.10.015.  Google Scholar

[36]

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. Google Scholar

[37]

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.  Google Scholar

[38]

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

[39]

R. Racke and B. Said-Houari, Global well-posedness of the Cauchy problem for the Jordan-Moore-Gibson-Thompson equation, preprint, http://nbn-resolving.de/urn:nbn:de:bsz:352-2-8ztzhsco3jj82 Google Scholar

[40]

M. A. Rammaha, Finite-time blow-up for nonlinear wave equations in high dimensions, Comm. Partial Differential Equations, 12 (1987), 677-700.  doi: 10.1080/03605308708820506.  Google Scholar

[41]

J. Schaeffer, Finite-time blow-up for $u_tt-\Delta u = H(u_r, u_t)$, Comm. Partial Differential Equations, 11 (1986), 513-543.  doi: 10.1080/03605308608820434.  Google Scholar

[42]

T. C. Sideris, Global behavior of solutions to nonlinear wave equations in three dimensions, Comm. Partial Differential Equations, 8 (1983), 1291-1323.  doi: 10.1080/03605308308820304.  Google Scholar

[43]

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.  Google Scholar

[44]

H. Takamura and K. Wakasa, Almost global solutions of semilinear wave equations with the critical exponent in high dimensions, Nonlinear Anal., 109 (2014), 187-229.  doi: 10.1016/j.na.2014.06.007.  Google Scholar

[45]

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

[46]

N. Tzvetkov, Existence of global solutions to nonlinear massless Dirac system and wave equation with small data, Tsukuba J. Math., 22 (1998), 193-211.  doi: 10.21099/tkbjm/1496163480.  Google Scholar

[47]

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.  Google Scholar

[48]

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.  Google Scholar

[49]

Y. Zhou, Blow up of solutions to the Cauchy problem for nonlinear wave equations, Chinese Ann. Math. Ser. B, 22 (2001), 275-280.  doi: 10.1142/S0252959901000280.  Google Scholar

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