# American Institute of Mathematical Sciences

doi: 10.3934/eect.2020074

## Decay rates for the moore-gibson-thompson equation with memory

 1 Ferhat Abbas university, Setif, 19000, Algeria 2 Department of Mathematics, College of Sciences, University of Sharjah, P. O. Box: 27272, Sharjah, United Arab Emirates

* Corresponding author: Belkacem Said Houari, bhouari@sharjah.ac.ae

Received  August 2019 Revised  February 2020 Published  June 2020

The main goal of this paper is to investigate the existence and stability of the solutions for the Moore–Gibson–Thompson equation (MGT) with a memory term in the whole spaces $\mathbb{R}^{N}$. The MGT equation arises from modeling high-frequency ultrasound waves as an alternative model to the well-known Kuznetsov's equation. First, following [8] and [26], we show that the problem is well-posed under an appropriate assumption on the coefficients of the system. Then, we built some Lyapunov functionals by using the energy method in Fourier space. These functionals allows us to get control estimates on the Fourier image of the solution. These estimates of the Fourier image together with some integral inequalities lead to the decay rate of the $L^{2}$-norm of the solution. We use two types of memory term here: type Ⅰ memory term and type Ⅲ memory term. Decay rates are obtained in both types. More precisely, decay rates of the solution are obtained depending on the exponential or polynomial decay of the memory kernel. More importantly, we show stability of the solution in both cases: a subcritical range of the parameters and a critical range. However for the type Ⅰ memory we show in the critical case that the solution has the regularity-loss property.

Citation: Hizia Bounadja, Belkacem Said Houari. Decay rates for the moore-gibson-thompson equation with memory. Evolution Equations & Control Theory, doi: 10.3934/eect.2020074
##### References:
 [1] M. O. Alves, A. H. Caixeta, M. A. Jorge Silva and J. H. Rodrigues, Moore–Gibson–Thompson equation with memory in a history framework: a semigroup approach, Z. Angew. Math. Phys., 69 (2018), Paper No. 106, 19 pp. doi: 10.1007/s00033-018-0999-5.  Google Scholar [2] M. S. Alves, C. Buriol, M. V. Ferreira, J. E. Muñoz Rivera, M. Sepúlveda and O. Vera, Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect, J. Math. Anal. Appl., 399 (2013), 472-479.  doi: 10.1016/j.jmaa.2012.10.019.  Google Scholar [3] J. A. Conejero, C. Lizama and F. Ródenas, Chaotic behaviour of the solutions of the Moore-Gibson-Thompson equation, Appl. Math. and Inf. Sciences., 9 (2015), 2233-2238.   Google Scholar [4] M. Conti, S. Gatti and V. Patta, Decay rates of Volterra equations on $\mathbb{R}^{N}$, Central European Journal of Mathematics, 5 (2007), 720-732.  doi: 10.2478/s11533-007-0024-2.  Google Scholar [5] F. Coulouvrat, On the equations of nonlinear acoustics, J. Acoustique, 5 (1992), 321-359.   Google Scholar [6] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rtion. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  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] 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 [9] 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 [10] A. Guessmia and S. A. Messaoudi, A new approach to the stability of an abstract system in the presence of infinite history, J. Math. Anal. Appl., 416 (2014), 212-228.  doi: 10.1016/j.jmaa.2014.02.030.  Google Scholar [11] P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete and Continuous Dynamical Systems B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189.  Google Scholar [12] B. Kaltenbacher, I. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore–Gibson–Thompson equation arising in high intensity ultrasound, Control and Cybernetics., 40 (2011), 971-988.   Google Scholar [13] B. Kaltenbacher, I. Lasiecka and M. K. Pospieszalska, Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 1250035(1-34). doi: 10.1142/S0218202512500352.  Google Scholar [14] B. Kaltenbacher, Mathematics of non linear acoustics, Evol Equ Control Theory., 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447.  Google Scholar [15] V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467. Google Scholar [16] 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 [17] I. Lasiecka and X. Wang, Moore–Gibson–Thompson equation with memory, part Ⅰ: exponential decay of energy, Z. Angew. Math. Phys., 67 (2016), Art. 17, 23 pp. doi: 10.1007/s00033-015-0597-8.  Google Scholar [18] I. Lasiecka and X. Wang, Moore–Gibson–Thompson equation with memory, part Ⅱ: General decay of energy, J. Differerential Equations., 259 (2015), 7610-7635.  doi: 10.1016/j.jde.2015.08.052.  Google Scholar [19] W. Liu, Z. Chen and D. Chen, New general decay results for a Moore–Gibson–Thompson equation with memory, Applicable Analysis, 2019, 1–20. doi: 10.1080/00036811.2019.1577390.  Google Scholar [20] E. Mainini and G. Mola, Exponential and polynomial decay for first order linear volterra evolution equations, Postdoctoral Fellowship of the Japan Society for the promotion of Sciences, 67 (2009), 93-111.  doi: 10.1090/S0033-569X-09-01145-X.  Google Scholar [21] 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 [22] S. A. Messaoudi and W. Al-Khulaifi, General and optimal decay for a quasilinear viscoelastic equation, Applied Mathematics Letters (ELSEVIER), 66 (2017), 16-22.  doi: 10.1016/j.aml.2016.11.002.  Google Scholar [23] V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.  doi: 10.1007/s00032-009-0098-3.  Google Scholar [24] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [25] M. Pellicer and J. Solà-Morales, Optimal scalar products in the Moore–Gibson–Thompson equation, Evol. Eqs. and Control Theory, 8 (2019), 203-220.  doi: 10.3934/eect.2019011.  Google Scholar [26] 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), 2017,447–478, https://arXiv.org/abs/1603.04270. doi: 10.1007/s00245-017-9471-8.  Google Scholar [27] M. Pellicer and B. Said-Houari, On the Cauchy problem for the standard linear solid model with heat conduction: Fourier versus Cattaneo, math. Appl., 2019, 1–39, https://arXiv.org/abs/1903.10181. Google Scholar [28] R. Racke and B. Said-Houari, Decay Rates for semilinear viscoelastic system in weighted spaces, Journal of Hyperbolic Differential Equations, 9 (2012), 67-103.  doi: 10.1142/S0219891612500026.  Google Scholar [29] R. Racke and B. Said-Houari, Global well-posedness of the Cauchy problem for the Jordan–Moore–Gibson–Thompson equation, Konstanzer Schriften in Mathematik, 382 (2019), 127. Google Scholar [30] B. E. Treeby, J. Jiri, B. T. R. Alistair and P. Cox, Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method, The Journal of the Acoustical Society of America, 131 (2012), 4324-4336.  doi: 10.1121/1.4712021.  Google Scholar

show all references

##### References:
 [1] M. O. Alves, A. H. Caixeta, M. A. Jorge Silva and J. H. Rodrigues, Moore–Gibson–Thompson equation with memory in a history framework: a semigroup approach, Z. Angew. Math. Phys., 69 (2018), Paper No. 106, 19 pp. doi: 10.1007/s00033-018-0999-5.  Google Scholar [2] M. S. Alves, C. Buriol, M. V. Ferreira, J. E. Muñoz Rivera, M. Sepúlveda and O. Vera, Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect, J. Math. Anal. Appl., 399 (2013), 472-479.  doi: 10.1016/j.jmaa.2012.10.019.  Google Scholar [3] J. A. Conejero, C. Lizama and F. Ródenas, Chaotic behaviour of the solutions of the Moore-Gibson-Thompson equation, Appl. Math. and Inf. Sciences., 9 (2015), 2233-2238.   Google Scholar [4] M. Conti, S. Gatti and V. Patta, Decay rates of Volterra equations on $\mathbb{R}^{N}$, Central European Journal of Mathematics, 5 (2007), 720-732.  doi: 10.2478/s11533-007-0024-2.  Google Scholar [5] F. Coulouvrat, On the equations of nonlinear acoustics, J. Acoustique, 5 (1992), 321-359.   Google Scholar [6] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rtion. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  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] 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 [9] 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 [10] A. Guessmia and S. A. Messaoudi, A new approach to the stability of an abstract system in the presence of infinite history, J. Math. Anal. Appl., 416 (2014), 212-228.  doi: 10.1016/j.jmaa.2014.02.030.  Google Scholar [11] P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete and Continuous Dynamical Systems B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189.  Google Scholar [12] B. Kaltenbacher, I. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore–Gibson–Thompson equation arising in high intensity ultrasound, Control and Cybernetics., 40 (2011), 971-988.   Google Scholar [13] B. Kaltenbacher, I. Lasiecka and M. K. Pospieszalska, Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 1250035(1-34). doi: 10.1142/S0218202512500352.  Google Scholar [14] B. Kaltenbacher, Mathematics of non linear acoustics, Evol Equ Control Theory., 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447.  Google Scholar [15] V. P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust., 16 (1971), 467. Google Scholar [16] 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 [17] I. Lasiecka and X. Wang, Moore–Gibson–Thompson equation with memory, part Ⅰ: exponential decay of energy, Z. Angew. Math. Phys., 67 (2016), Art. 17, 23 pp. doi: 10.1007/s00033-015-0597-8.  Google Scholar [18] I. Lasiecka and X. Wang, Moore–Gibson–Thompson equation with memory, part Ⅱ: General decay of energy, J. Differerential Equations., 259 (2015), 7610-7635.  doi: 10.1016/j.jde.2015.08.052.  Google Scholar [19] W. Liu, Z. Chen and D. Chen, New general decay results for a Moore–Gibson–Thompson equation with memory, Applicable Analysis, 2019, 1–20. doi: 10.1080/00036811.2019.1577390.  Google Scholar [20] E. Mainini and G. Mola, Exponential and polynomial decay for first order linear volterra evolution equations, Postdoctoral Fellowship of the Japan Society for the promotion of Sciences, 67 (2009), 93-111.  doi: 10.1090/S0033-569X-09-01145-X.  Google Scholar [21] 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 [22] S. A. Messaoudi and W. Al-Khulaifi, General and optimal decay for a quasilinear viscoelastic equation, Applied Mathematics Letters (ELSEVIER), 66 (2017), 16-22.  doi: 10.1016/j.aml.2016.11.002.  Google Scholar [23] V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.  doi: 10.1007/s00032-009-0098-3.  Google Scholar [24] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [25] M. Pellicer and J. Solà-Morales, Optimal scalar products in the Moore–Gibson–Thompson equation, Evol. Eqs. and Control Theory, 8 (2019), 203-220.  doi: 10.3934/eect.2019011.  Google Scholar [26] 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), 2017,447–478, https://arXiv.org/abs/1603.04270. doi: 10.1007/s00245-017-9471-8.  Google Scholar [27] M. Pellicer and B. Said-Houari, On the Cauchy problem for the standard linear solid model with heat conduction: Fourier versus Cattaneo, math. Appl., 2019, 1–39, https://arXiv.org/abs/1903.10181. Google Scholar [28] R. Racke and B. Said-Houari, Decay Rates for semilinear viscoelastic system in weighted spaces, Journal of Hyperbolic Differential Equations, 9 (2012), 67-103.  doi: 10.1142/S0219891612500026.  Google Scholar [29] R. Racke and B. Said-Houari, Global well-posedness of the Cauchy problem for the Jordan–Moore–Gibson–Thompson equation, Konstanzer Schriften in Mathematik, 382 (2019), 127. Google Scholar [30] B. E. Treeby, J. Jiri, B. T. R. Alistair and P. Cox, Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method, The Journal of the Acoustical Society of America, 131 (2012), 4324-4336.  doi: 10.1121/1.4712021.  Google Scholar
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