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January  2018, 17(1): 243-265. doi: 10.3934/cpaa.2018015

## Hölder regularity for the Moore-Gibson-Thompson equation with infinite delay

 1 Universidad de Santiago de Chile, Departamento de Matemáticay Ciencia de la Computación, Las Sophoras 173, Estación Central, Santiago, Chile 2 BCAM-Basque Center for Applied Mathematics, Mazarredo, 14, E48009 Bilbao, Basque Country, Spain

* Corresponding author

Received  December 2016 Revised  June 2017 Published  September 2017

Fund Project: The first author is supported by the Project POSTDOC DICYT-041633LY at the USACH. The second author is partially supported by CONICYT, under Fondecyt Grant number 1140258 and and CONICYT - PIA - Anillo ACT1416. The third author is supported by the Basque Government through the BERC 2014-2017 program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and GEAGAM, 644202 H2020-MSCA-RISE-2014.

We characterize the well-posedness of a third order in time equation with infinite delay in Hölder spaces, solely in terms of spectral properties concerning the data of the problem. Our analysis includes the case of the linearized Kuznetzov and Westerwelt equations. We show in case of the Laplacian operator the new and surprising fact that for the standard memory kernel $g(t)=\frac{t^{ν-1}}{Γ(ν)}e^{-at}$ the third order problem is ill-posed whenever $0<ν ≤q 1$ and $a$ is inversely proportional to one of the terms of the given model.

Citation: Luciano Abadías, Carlos Lizama, Marina Murillo-Arcila. Hölder regularity for the Moore-Gibson-Thompson equation with infinite delay. Communications on Pure & Applied Analysis, 2018, 17 (1) : 243-265. doi: 10.3934/cpaa.2018015
##### References:
 [1] F. Alabau-Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342-1372. doi: 10.1016/j.jfa.2007.09.012. Google Scholar [2] D. Araya and C. Lizama, Existence of asymptotically almost automorphic solutions for a third order differential equation, E.J. Qualitative Theory of Diff. Eq., 53 (2012), 1-20. Google Scholar [3] W. Arendt, C. J. Batty and S. Bu, Fourier multipliers for Hölder continuous functions and maximal regularity, Studia Math., 160 (2004), 23-51. doi: 10.4064/sm160-1-2. Google Scholar [4] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems Second edition, Monographs in Mathematics. 96 Birkhäuser, 2011. doi: 10.1007/978-3-0348-0087-7. Google Scholar [5] H. Brézis, Analyse Fonctionnelle Masson, Paris, 1983. Google Scholar [6] S. Bu, Well-posedness of second order degenerate differential equations in vector-valued function spaces, Studia Math., 214 (2013), 1-16. doi: 10.4064/sm214-1-1. Google Scholar [7] S. Bu and Y. Fang, Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces, Studia Math., 184 (2008), 103-119. doi: 10.4064/sm184-2-1. Google Scholar [8] S. Bu and G. Cai, Well-posedness of second order degenerate differential equations in Hölder continuous function spaces, Expo. Math., 34 (2016), 223-236. doi: 10.1016/j.exmath.2015.07.003. Google Scholar [9] G. Cai and S. Bu, Periodic solutions of third-order degenerate differential equations in vector-valued functional spaces, Israel J. Math., 212 (2016), 163-188. doi: 10.1007/s11856-016-1282-0. Google Scholar [10] G. Cai and S. Bu, Well-posedness of second order degenerate integrodifferential equations with infinite delay in vector-valued function spaces, Math. Nachr., 289 (2016), 436-451. doi: 10.1002/mana.201400112. Google Scholar [11] A. H. Caixeta, I. Lasiecka and V. N. D. Cavalcanti, Global attractors for a third order in time nonlinear dynamics, J. Differential Equations, 261 (2016), 113-147. doi: 10.1016/j.jde.2016.03.006. Google Scholar [12] R. Chill and S. Srivastava, $L_p$-maximal regularity for second order Cauchy problems, Math. Z., 251 (2005), 751-781. doi: 10.1007/s00209-005-0815-8. Google Scholar [13] Ph. Clément and G. Da Prato, Existence and regularity results for an integral equation with infinite delay in a Banach space, Integral Equations and Operator Theory, 11 (1988), 480-500. doi: 10.1007/BF01199303. Google Scholar [14] A. Conejero, C. Lizama and F. Rodenas, Chaotic behaviour of the solutions of the Moore-Gibson-Thomson equation, Appl. Math. Inf. Sci., 9 (2015), 1-16. Google Scholar [15] C. Cuevas and C. Lizama, Well-posedness for a class of flexible structure in Hölder spaces, Math. Probl. Eng., vol, 2009 (2009), 1-13. doi: 10.1155/2009/358329. Google Scholar [16] G. Da Prato and E. Sinestrari, Hölder regularity for non autonomous abstract parabolic equations, Israel J. Math., 42 (1982), 1-19. doi: 10.1007/BF02765006. Google Scholar [17] 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 [18] F. Dell'oro and V. Pata, On the Moore-Gibson-Thompson equation and its relation to linear viscoelasticity, Appl. Math. Optim. (2016). doi: 10.1007/s00245-016-9365-1.Google Scholar [19] B. De Andrade and C. Lizama, Existence of asymptotically almost periodic solutions for damped wave equations, J. Math. Anal. Appl., 382 (2011), 761-771. doi: 10.1016/j.jmaa.2011.04.078. Google Scholar [20] R. Denk, M. Hieber and J. Prüss, $R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type Mem. Amer. Math. Soc. 166 (2003), no. 788. doi: 10.1090/memo/0788. Google Scholar [21] R. Denk, J. Prüss and R. Zacher, Maximal $Lp$-regularity of parabolic problems with boundary dynamics of relaxation type, J. Funct. Anal., 255 (2008), 3149-3187. doi: 10.1016/j.jfa.2008.07.012. Google Scholar [22] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations Graduate texts in Mathematics. 194 Springer, New York, 2000. Google Scholar [23] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces North-Holland Mathematics Studies, 108, North-Holland, Amsterdam, 1985. Google Scholar [24] C. Fernández, C. Lizama and V. Poblete, Maximal regularity for flexible structural systems in Lebesgue spaces, Math. Probl. Eng., vol, 2010 (2010), 1-5. doi: 10.1155/2010/196956. Google Scholar [25] C. Fernández, C. Lizama and V. Poblete, Regularity of solutions for a third order differential equation in Hilbert spaces, Appl. Math. Comput., 217 (2011), 8522-8533. doi: 10.1016/j.amc.2011.03.056. Google Scholar [26] G. 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 [27] M. Haase. The Functional Calculus for Sectorial Operators Operator Theory: Advances and Applications, 169. Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8. Google Scholar [28] B. Kaltenbacher, I. Lasiecka and R. Marchand, Well-posedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control Cybernet., 40 (2011), 971-988. Google Scholar [29] 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), 1-34. doi: 10.1142/S0218202512500352. Google Scholar [30] B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, Discrete Contin. Dyn. Syst. 2011, Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl. Vol. Ⅱ, 763-773 Google Scholar [31] V. Keyantuo and C. Lizama, Hölder continuous solutions for integro-differential equations and maximal regularity, J. Differential Equations, 230 (2006), 634-660. doi: 10.1016/j.jde.2006.07.018. Google Scholar [32] N. T. Lan, On the nonautonomous higher-order Cauchy problems, Differential Integral Equations, 14 (2001), 241-256. Google Scholar [33] I. Lasiecka and X. Wang, Moore-Gibson-Thompson equation with memory, part I: Exponential decay energy, Z. Angew. Math. Phys., (2016), 67-17. doi: 10.1007/s00033-015-0597-8. Google Scholar [34] 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 [35] R. Marchand, T. McDevitt and R. Triggiani, An abstract semigroup approach to the third-order Moore-Gibson-Thomson 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 [36] S. A. Messaoudi and S.E. Mukiana, Existence and decay solutions to a viscoelastic plate equation, Electr. J. Diff. Equ., 22 (2016), 1-14. Google Scholar [37] F. Neubrander, Well-posedness of higher order abstract Cauchy problems, Trans. Amer. Math. Soc., 295 (1986), 257-290. doi: 10.2307/2000156. Google Scholar [38] J. Prüss, Evolutionary Integral Equations and Applications Monographs in Math. 87 Birkhäuser, 1993. doi: 10.1007/978-3-0348-8570-6. Google Scholar [39] J. Prüss, Decay properties for the solutions of a partial differential equation with memory, Arch. Math., 92 (2009), 158-173. doi: 10.1007/s00013-008-2936-x. Google Scholar [40] T. J. Xiao and J. Liang, The Cauchy Problem for Higher-order Abstract Differential Equations Lecture Notes in Mathematics, 1701. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-540-49479-9. Google Scholar

show all references

##### References:
 [1] F. Alabau-Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342-1372. doi: 10.1016/j.jfa.2007.09.012. Google Scholar [2] D. Araya and C. Lizama, Existence of asymptotically almost automorphic solutions for a third order differential equation, E.J. Qualitative Theory of Diff. Eq., 53 (2012), 1-20. Google Scholar [3] W. Arendt, C. J. Batty and S. Bu, Fourier multipliers for Hölder continuous functions and maximal regularity, Studia Math., 160 (2004), 23-51. doi: 10.4064/sm160-1-2. Google Scholar [4] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems Second edition, Monographs in Mathematics. 96 Birkhäuser, 2011. doi: 10.1007/978-3-0348-0087-7. Google Scholar [5] H. Brézis, Analyse Fonctionnelle Masson, Paris, 1983. Google Scholar [6] S. Bu, Well-posedness of second order degenerate differential equations in vector-valued function spaces, Studia Math., 214 (2013), 1-16. doi: 10.4064/sm214-1-1. Google Scholar [7] S. Bu and Y. Fang, Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces, Studia Math., 184 (2008), 103-119. doi: 10.4064/sm184-2-1. Google Scholar [8] S. Bu and G. Cai, Well-posedness of second order degenerate differential equations in Hölder continuous function spaces, Expo. Math., 34 (2016), 223-236. doi: 10.1016/j.exmath.2015.07.003. Google Scholar [9] G. Cai and S. Bu, Periodic solutions of third-order degenerate differential equations in vector-valued functional spaces, Israel J. Math., 212 (2016), 163-188. doi: 10.1007/s11856-016-1282-0. Google Scholar [10] G. Cai and S. Bu, Well-posedness of second order degenerate integrodifferential equations with infinite delay in vector-valued function spaces, Math. Nachr., 289 (2016), 436-451. doi: 10.1002/mana.201400112. Google Scholar [11] A. H. Caixeta, I. Lasiecka and V. N. D. Cavalcanti, Global attractors for a third order in time nonlinear dynamics, J. Differential Equations, 261 (2016), 113-147. doi: 10.1016/j.jde.2016.03.006. Google Scholar [12] R. Chill and S. Srivastava, $L_p$-maximal regularity for second order Cauchy problems, Math. Z., 251 (2005), 751-781. doi: 10.1007/s00209-005-0815-8. Google Scholar [13] Ph. Clément and G. Da Prato, Existence and regularity results for an integral equation with infinite delay in a Banach space, Integral Equations and Operator Theory, 11 (1988), 480-500. doi: 10.1007/BF01199303. Google Scholar [14] A. Conejero, C. Lizama and F. Rodenas, Chaotic behaviour of the solutions of the Moore-Gibson-Thomson equation, Appl. Math. Inf. Sci., 9 (2015), 1-16. Google Scholar [15] C. Cuevas and C. Lizama, Well-posedness for a class of flexible structure in Hölder spaces, Math. Probl. Eng., vol, 2009 (2009), 1-13. doi: 10.1155/2009/358329. Google Scholar [16] G. Da Prato and E. Sinestrari, Hölder regularity for non autonomous abstract parabolic equations, Israel J. Math., 42 (1982), 1-19. doi: 10.1007/BF02765006. Google Scholar [17] 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 [18] F. Dell'oro and V. Pata, On the Moore-Gibson-Thompson equation and its relation to linear viscoelasticity, Appl. Math. Optim. (2016). doi: 10.1007/s00245-016-9365-1.Google Scholar [19] B. De Andrade and C. Lizama, Existence of asymptotically almost periodic solutions for damped wave equations, J. Math. Anal. Appl., 382 (2011), 761-771. doi: 10.1016/j.jmaa.2011.04.078. Google Scholar [20] R. Denk, M. Hieber and J. Prüss, $R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type Mem. Amer. Math. Soc. 166 (2003), no. 788. doi: 10.1090/memo/0788. Google Scholar [21] R. Denk, J. Prüss and R. Zacher, Maximal $Lp$-regularity of parabolic problems with boundary dynamics of relaxation type, J. Funct. Anal., 255 (2008), 3149-3187. doi: 10.1016/j.jfa.2008.07.012. Google Scholar [22] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations Graduate texts in Mathematics. 194 Springer, New York, 2000. Google Scholar [23] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces North-Holland Mathematics Studies, 108, North-Holland, Amsterdam, 1985. Google Scholar [24] C. Fernández, C. Lizama and V. Poblete, Maximal regularity for flexible structural systems in Lebesgue spaces, Math. Probl. Eng., vol, 2010 (2010), 1-5. doi: 10.1155/2010/196956. Google Scholar [25] C. Fernández, C. Lizama and V. Poblete, Regularity of solutions for a third order differential equation in Hilbert spaces, Appl. Math. Comput., 217 (2011), 8522-8533. doi: 10.1016/j.amc.2011.03.056. Google Scholar [26] G. 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 [27] M. Haase. The Functional Calculus for Sectorial Operators Operator Theory: Advances and Applications, 169. Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8. Google Scholar [28] B. Kaltenbacher, I. Lasiecka and R. Marchand, Well-posedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control Cybernet., 40 (2011), 971-988. Google Scholar [29] 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), 1-34. doi: 10.1142/S0218202512500352. Google Scholar [30] B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, Discrete Contin. Dyn. Syst. 2011, Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl. Vol. Ⅱ, 763-773 Google Scholar [31] V. Keyantuo and C. Lizama, Hölder continuous solutions for integro-differential equations and maximal regularity, J. Differential Equations, 230 (2006), 634-660. doi: 10.1016/j.jde.2006.07.018. Google Scholar [32] N. T. Lan, On the nonautonomous higher-order Cauchy problems, Differential Integral Equations, 14 (2001), 241-256. Google Scholar [33] I. Lasiecka and X. Wang, Moore-Gibson-Thompson equation with memory, part I: Exponential decay energy, Z. Angew. Math. Phys., (2016), 67-17. doi: 10.1007/s00033-015-0597-8. Google Scholar [34] 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 [35] R. Marchand, T. McDevitt and R. Triggiani, An abstract semigroup approach to the third-order Moore-Gibson-Thomson 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 [36] S. A. Messaoudi and S.E. Mukiana, Existence and decay solutions to a viscoelastic plate equation, Electr. J. Diff. Equ., 22 (2016), 1-14. Google Scholar [37] F. Neubrander, Well-posedness of higher order abstract Cauchy problems, Trans. Amer. Math. Soc., 295 (1986), 257-290. doi: 10.2307/2000156. Google Scholar [38] J. Prüss, Evolutionary Integral Equations and Applications Monographs in Math. 87 Birkhäuser, 1993. doi: 10.1007/978-3-0348-8570-6. Google Scholar [39] J. Prüss, Decay properties for the solutions of a partial differential equation with memory, Arch. Math., 92 (2009), 158-173. doi: 10.1007/s00013-008-2936-x. Google Scholar [40] T. J. Xiao and J. Liang, The Cauchy Problem for Higher-order Abstract Differential Equations Lecture Notes in Mathematics, 1701. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-540-49479-9. Google Scholar
Example of a parametric plot (${\mathfrak R}{\mathfrak e}\, \beta_2(\eta), {\mathfrak I}{\mathfrak m}\,\beta_2(\eta)$)
Example of a parametric plot (${\mathfrak R}{\mathfrak e}\, \beta_3(\eta), {\mathfrak I}{\mathfrak m}\,\beta_3(\eta)$)
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