December  2021, 10(4): 673-687. 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  December 2021 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.

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 and Control Theory, 2021, 10 (4) : 673-687. 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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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]

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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]

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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

[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.

[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.

[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.

[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.

[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.

[45]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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]

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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]

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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

[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.

[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.

[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.

[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.

[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.

[45]

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

[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.

[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.

[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.

[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.

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