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May  2022, 21(5): 1773-1792. doi: 10.3934/cpaa.2022046

Formation of singularities of solutions to the Cauchy problem for semilinear Moore-Gibson-Thompson equations

1. 

Department of Mathematics, North University of China, Taiyuan 030051, China

2. 

Department of Mathematics, Southwest Jiaotong University, Chengdu 611756, China

3. 

Data Science and Technology, North University of China, Taiyuan 030051, China

* Corresponding author

Received  October 2021 Revised  January 2022 Published  May 2022 Early access  February 2022

Fund Project: The project is supported by Natural Science Foundation of Shanxi Province of China (No. 201901D211276), Fundamental Research Program of Shanxi Province (No. 20210302123045) and National Natural Science Foundation of China (No. 11971394)

This paper is devoted to investigating formation of singularities for solutions to semilinear Moore-Gibson-Thompson equations with power type nonlinearity $ |u|^{p} $, derivative type nonlinearity $ |u_{t}|^{p} $ and combined type nonlinearities $ |u_{t}|^{p}+|u|^{q} $ in the case of single equation, combined type nonlinearities $ |v_{t}|^{p_{1}}+|v|^{q_{1}} $, $ |u_{t}|^{p_{2}}+|u|^{q_{2}} $, combined and power type nonlinearities $ |v_{t}|^{p_{1}}+|v|^{q_{1}} $, $ |u|^{q_{2}} $, combined and derivative type nonlinearities $ |v_{t}|^{p_{1}}+|v|^{q_{1}} $, $ |u_{t}|^{p_{2}} $ in the case of coupled system, respectively. More precisely, blow-up results of solutions to problems in the sub-critical and critical cases are derived by applying test function technique. Moreover, upper bound lifespan estimates of solutions to the coupled systems are investigated. The main new contribution is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent.

Citation: Sen Ming, Han Yang, Xiongmei Fan. Formation of singularities of solutions to the Cauchy problem for semilinear Moore-Gibson-Thompson equations. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1773-1792. doi: 10.3934/cpaa.2022046
References:
[1]

A. B. Aissa, Stabilization of the petrovsky wave nonlinear coupled system with strong damping, arXiv: 2012.07109v1.

[2]

F. Bucci and M. Eller, The Cauchy Dirichlet problem for the Moore-Gibson-Thompson equation, C. R. Math. Acad. Sci. Paris, 359 (2021), 881-903.  doi: 10.5802/crmath.231.

[3]

W. H. Chen and A. Z. Fino, Blow-up of solutions to semilinear strongly damped wave equations with different nonlinear terms in an exterior domain, Math. Methods Appl. Sci., 44 (2021), 6787-6807.  doi: 10.1002/mma.7223.

[4]

W. H. Chen and R. Ikehata, The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case, J. Differ. Equ., 292 (2021), 176-219.  doi: 10.1016/j.jde.2021.05.011.

[5]

W. H. Chen and A. Palmieri, A blow-up result for the semilinear Moore-Gibson-Thompson equation with nonlinearity of derivative type in the conservative case, Evol. Equ. Control Theory, 10 (2021), 673-687.  doi: 10.3934/eect.2020085.

[6]

W. H. Chen and A. Palmieri, Non-existence 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.

[7]

T. A. Dao, A result for non-existence of global solutions to semilinear structural damped wave model, arXiv: 1912.07066v1.

[8]

T. A. Dao and A. Z. Fino, Critical exponent for semilinear structurally damped wave equation of derivative type, Math. Methods Appl. Sci., 43 (2020), 9766-9775.  doi: 10.1002/mma.6649.

[9]

A. Z. Fino, Finite time blow-up for wave equations with strong damping in an exterior domain, Mediterr. J. Math., 17 (2020), 21 pp. doi: 10.1007/s00009-020-01607-2.

[10]

V. Georgiev, H. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, arXiv: math/9912206.

[11]

M. Hamouda and M. A. Hamza, New blow-up result for the weakly coupled wave equations with a scale invariant damping and time derivative nonlinearity, arXiv: 2008.06569v1.

[12]

K. HidanoC. B. 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.

[13]

M. IkedaM. Sobajima and K. Wakasa, Blow-up phenomena of semilinear wave equations and their weakly couples system, J. Differ. Equ., 267 (2019), 5165-5201.  doi: 10.1016/j.jde.2019.05.029.

[14]

M. IkedaT. Tanaka and K. Wakasa, Critical exponent for the wave equation with a time dependent scale invariant damping and a cubic convolution, J. Differ. Equ., 270 (2021), 916-946.  doi: 10.1016/j.jde.2020.08.047.

[15]

N. A. LaiN. M. Schiavone and H. Takamura, Heat like and wave like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma, J. Differ. Equ., 269 (2020), 11575-11620.  doi: 10.1016/j.jde.2020.08.020.

[16]

N. A. Lai and Z. H. Tu, Strauss exponent for semilinear wave equations with scsttering space dependent damping, J. Math. Anal. Appl., 489 (2020), 24 pp. doi: 10.1016/j.jmaa.2020.124189.

[17]

Y. H. Lin, N. A. Lai and S. Ming, Lifespan estimate for semilinear wave equation in Schwarzschild spacetime, Appl. Math. Lett., 99 (2020), 4 pp. doi: 10.1016/j.aml.2019.105997.

[18]

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.

[19]

S. Ming, S. Y. Lai and X. M. Fan, Lifespan estimates of solutions to quasilinear wave equations with scattering damping, J. Math. Anal. Appl., 492 (2020), 15 pp. doi: 10.1016/j.jmaa.2020.124441.

[20]

S. MingS. Y. Lai and X. M. Fan, Blow-up for a coupled system of semilinear wave equations with scattering dampings and combined nonlinearities, Appl. Anal., (2020), 1-21.  doi: 10.1080/00036811.2020.1834086.

[21]

S. MingH. Yang and X. M. Fan, Blow-up and lifespan estimates of solutions to the weakly coupled system of semilinear Moore-Gibson-Thompson equations, Math. Methods Appl. Sci., 44 (2021), 10972-10992.  doi: 10.1002/mma.7462.

[22]

A. Palmieri and H. Takamura, Non-existence 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), 20 pp. doi: 10.1007/s00009-019-1445-4.

[23]

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.

[24]

A. Palmieri and H. Takamura, Non-existence of global solutions for a weakly coupled system of semilinear damped wave equations in the scattering case with mixed nonlinear terms, NoDEA Nonlinear Differ. Equ. Appl., 27 (2020), 39 pp. doi: 10.1007/s00030-020-00662-8.

[25]

M. Pellicer and B. Said-Houari, Well posedness 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.

[26]

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

[27]

K. Wakasa and B. Yordanov, Blow-up of solutions to critical semilinear wave equations with variable coefficients, J. Differ. Equ., 266 (2019), 5360-5376.  doi: 10.1016/j.jde.2018.10.028.

[28]

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.

[29]

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

[30]

Y. Zhou and W. Han, Lifespan of solutions to critical semilinear wave equations, Commun. Partial Differ. Equ., 39 (2014), 439-451.  doi: 10.1080/03605302.2013.863914.

show all references

References:
[1]

A. B. Aissa, Stabilization of the petrovsky wave nonlinear coupled system with strong damping, arXiv: 2012.07109v1.

[2]

F. Bucci and M. Eller, The Cauchy Dirichlet problem for the Moore-Gibson-Thompson equation, C. R. Math. Acad. Sci. Paris, 359 (2021), 881-903.  doi: 10.5802/crmath.231.

[3]

W. H. Chen and A. Z. Fino, Blow-up of solutions to semilinear strongly damped wave equations with different nonlinear terms in an exterior domain, Math. Methods Appl. Sci., 44 (2021), 6787-6807.  doi: 10.1002/mma.7223.

[4]

W. H. Chen and R. Ikehata, The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case, J. Differ. Equ., 292 (2021), 176-219.  doi: 10.1016/j.jde.2021.05.011.

[5]

W. H. Chen and A. Palmieri, A blow-up result for the semilinear Moore-Gibson-Thompson equation with nonlinearity of derivative type in the conservative case, Evol. Equ. Control Theory, 10 (2021), 673-687.  doi: 10.3934/eect.2020085.

[6]

W. H. Chen and A. Palmieri, Non-existence 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.

[7]

T. A. Dao, A result for non-existence of global solutions to semilinear structural damped wave model, arXiv: 1912.07066v1.

[8]

T. A. Dao and A. Z. Fino, Critical exponent for semilinear structurally damped wave equation of derivative type, Math. Methods Appl. Sci., 43 (2020), 9766-9775.  doi: 10.1002/mma.6649.

[9]

A. Z. Fino, Finite time blow-up for wave equations with strong damping in an exterior domain, Mediterr. J. Math., 17 (2020), 21 pp. doi: 10.1007/s00009-020-01607-2.

[10]

V. Georgiev, H. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, arXiv: math/9912206.

[11]

M. Hamouda and M. A. Hamza, New blow-up result for the weakly coupled wave equations with a scale invariant damping and time derivative nonlinearity, arXiv: 2008.06569v1.

[12]

K. HidanoC. B. 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.

[13]

M. IkedaM. Sobajima and K. Wakasa, Blow-up phenomena of semilinear wave equations and their weakly couples system, J. Differ. Equ., 267 (2019), 5165-5201.  doi: 10.1016/j.jde.2019.05.029.

[14]

M. IkedaT. Tanaka and K. Wakasa, Critical exponent for the wave equation with a time dependent scale invariant damping and a cubic convolution, J. Differ. Equ., 270 (2021), 916-946.  doi: 10.1016/j.jde.2020.08.047.

[15]

N. A. LaiN. M. Schiavone and H. Takamura, Heat like and wave like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma, J. Differ. Equ., 269 (2020), 11575-11620.  doi: 10.1016/j.jde.2020.08.020.

[16]

N. A. Lai and Z. H. Tu, Strauss exponent for semilinear wave equations with scsttering space dependent damping, J. Math. Anal. Appl., 489 (2020), 24 pp. doi: 10.1016/j.jmaa.2020.124189.

[17]

Y. H. Lin, N. A. Lai and S. Ming, Lifespan estimate for semilinear wave equation in Schwarzschild spacetime, Appl. Math. Lett., 99 (2020), 4 pp. doi: 10.1016/j.aml.2019.105997.

[18]

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.

[19]

S. Ming, S. Y. Lai and X. M. Fan, Lifespan estimates of solutions to quasilinear wave equations with scattering damping, J. Math. Anal. Appl., 492 (2020), 15 pp. doi: 10.1016/j.jmaa.2020.124441.

[20]

S. MingS. Y. Lai and X. M. Fan, Blow-up for a coupled system of semilinear wave equations with scattering dampings and combined nonlinearities, Appl. Anal., (2020), 1-21.  doi: 10.1080/00036811.2020.1834086.

[21]

S. MingH. Yang and X. M. Fan, Blow-up and lifespan estimates of solutions to the weakly coupled system of semilinear Moore-Gibson-Thompson equations, Math. Methods Appl. Sci., 44 (2021), 10972-10992.  doi: 10.1002/mma.7462.

[22]

A. Palmieri and H. Takamura, Non-existence 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), 20 pp. doi: 10.1007/s00009-019-1445-4.

[23]

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.

[24]

A. Palmieri and H. Takamura, Non-existence of global solutions for a weakly coupled system of semilinear damped wave equations in the scattering case with mixed nonlinear terms, NoDEA Nonlinear Differ. Equ. Appl., 27 (2020), 39 pp. doi: 10.1007/s00030-020-00662-8.

[25]

M. Pellicer and B. Said-Houari, Well posedness 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.

[26]

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

[27]

K. Wakasa and B. Yordanov, Blow-up of solutions to critical semilinear wave equations with variable coefficients, J. Differ. Equ., 266 (2019), 5360-5376.  doi: 10.1016/j.jde.2018.10.028.

[28]

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.

[29]

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

[30]

Y. Zhou and W. Han, Lifespan of solutions to critical semilinear wave equations, Commun. Partial Differ. Equ., 39 (2014), 439-451.  doi: 10.1080/03605302.2013.863914.

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