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

Wenhui Chen; Alessandro Palmieri

doi:10.3934/eect.2020085

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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

Wenhui Chen ^1,, and Alessandro Palmieri ^2,

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 Early access 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.

Keywords: Moore – Gibson – Thompson equation, semilinear third order equation, blow – up, lifespan estimates, Glassey exponent.

Mathematics Subject Classification: Primary: 35B44, 35L30; Secondary: 35L76, 35B33, 35L25.

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. Agemi, Y. 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. Caixeta, I. 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'Oro, I. 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. Hidano, C. 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. Kaltenbacher, I. 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. Lai, H. 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. Marchand, T. 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. Agemi, Y. 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. Caixeta, I. 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'Oro, I. 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. Hidano, C. 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. Kaltenbacher, I. 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. Lai, H. 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. Marchand, T. 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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American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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