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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 |
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.
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. 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. |
[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. 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. |
[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'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. |
[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. 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. |
[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. 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.
|
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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'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. |
[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. 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. |
[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. 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.
|
[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. 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. |
[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. 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. |
[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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