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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 & Continuous Dynamical Systems - A, 2020, 40 (9) : 5513-5540. doi: 10.3934/dcds.2020236
##### References:
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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.  Google Scholar [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.  Google Scholar [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.  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. 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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [43] P. A. Thompson, Compressible Fluid Dynamics, McGraw-Hill, New York, 1972. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### References:
 [1] 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 [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.  Google Scholar [3] 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 [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. Google Scholar [5] F. Dell'Oro, I. Lasiecka and V. Pata, On the MGT equation with memory of type Ⅱ, preprint, arXiv: 1904.08203. Google Scholar [6] 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 [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.  Google Scholar [8] P. M. N. Dharmawardane, J. 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.  Google Scholar [9] S. Di Pomponio and V. Georgiev, Life-span of subcritical semilinear wave equation, Asymptot. Anal., 28 (2001), 91-114.   Google Scholar [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.  Google Scholar [11] V. Georgiev, H. 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.  Google Scholar [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.  Google Scholar [13] R. T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.  doi: 10.1007/BF01162066.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math., 33 (1980), 501-505.  doi: 10.1002/cpa.3160330403.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [43] P. A. Thompson, Compressible Fluid Dynamics, McGraw-Hill, New York, 1972. Google Scholar [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.  Google Scholar [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.  Google Scholar [46] B. T. 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