Global existence and general decay of Moore–Gibson–Thompson equation with not necessarily decreasing kernel
Resumen
In this paper, we consider the Moore.Gibson.Thompson equations. By using the potential well theory we obtain the existence of a global solution. Then, we prove the general decay result of solutions under weaker assumptions than the ones frequently used in the literature. In particular, the kernels we are considering are not necessarily exponentially decaying to zero as was assumed before. The present results improve also a previous work of the authors.
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Alabau-Boussouira, F., A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems. J Differ. Equ. 248, 1473 − 1517, (2010). https://doi.org/10.1016/j.jde.2009.12.005
Boumaza, N., Boulaaras, S.: Ggeneral decay for Kirchhoff type in viscoelasticity with not necessarily decreasing kernel. Math Meth Appl Sci. 1-20, (2018). https://doi.org/10.1002/mma.5117
Boulaaras, S., Draifia, A., Alnegga, M.: Polynomial decay rate for kirchhoff type in viscoelasticity with logarithmic nonlinearity and not necessarily decreasing kernel. Symmetry 11,226, (2019); https://doi.org/10.3390/sym11020226
Boulaaras, S., Zara¨ı, A., Draifia, A., Galerkin method for nonlocal mixed boundary value problem for the Moore-GibsonThompson equation with integral condition. Math Meth Appl Sci. 1 − 16, (2019) https://doi.org/10.1002/mma.5540
Boulaaras, S., Draifia, A., Zennir, K., General decay of nonlinear viscoelastic Kirchhoff equation with BalakrishnanTaylor damping and logarithmic nonlinearity. Math Meth Appl Sci. 1 − 20, (2019). https://doi.org/10.1002/mma.5693
Cavalcanti, M. M., Cavalcanti, A. D. D., Lasiecka, I.,Wang, X., Existence and sharp decay rate estimates for a von Karman system with long memory. Nonlinear Anal. Real World Appl. 22, 289 − 306, (2015). https://doi.org/10.1016/j.nonrwa.2014.09.016
Cavalcanti, M. M., Cavalcanti, V. N. D., Martinez, P., General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal. 68(1), 177 − 193, (2008). https://doi.org/10.1016/j.na.2006.10.040
Cavalcanti, M. M., Oquendo, H. P., Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310 − 1324, (2003). https://doi.org/10.1137/S0363012902408010
Dell'Oroa, F., LasieckabPata, I. V., Moore-Gibson-Thompson equation with memory in the critical case. Journal of Differential Equations · June (2016) https://doi.org/10.1016/j.jde.2016.06.025
Dafermos, C. M., Asymptotic stability in viscoelasticity. Arch. Rat. Mech. Anal. 37, 297 − 308, (1970). https://doi.org/10.1007/BF00251609
Fabrizio, M., Polidoro, S., Asymptotic decay for some differential systems with fading memory. Appl. Anal. 81(6), 1245−1264, (2002). https://doi.org/10.1080/0003681021000035588
Han, X., Wang, M.: General decay rates of energy for the second order evolutions equations with memory. Acta Appl. Math. 110, 195 − 207, (2010). https://doi.org/10.1007/s10440-008-9397-x
Hrusa, W. J., Nohel, J. A., Renardy, M., Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman (1987).
Jordan, P. M., Nonlinear acoustic phenomena in viscous thermally relaxing fluids: Shock bifurcation and the emergence of diffusive solitons. J. Acoust. Soc. Am. 124, 2491, (2008). https://doi.org/10.1121/1.4782790
Kaltenbacher, B., Lasiecka, I., Marchand, R., well-posedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound. Control Cybern. 40(4), 971 − 988, (2011).
Lasiecka, I., Wang, X., Moore-Gibson-Thompson equation with memory, part I: exponential decay of energy. Z. Angew. Math. Phys. 67, 17, (2016). https://doi.org/10.1007/s00033-015-0597-8
Lasiecka, I., Wang, X., Moore-Gibson-Thompson equation with memory, part II: general decay of energy. J. Differ. Equ. 259(12), 7610 − 7635, (2015). https://doi.org/10.1016/j.jde.2015.08.052
Lebon, G., Cloot, A.: Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics. Wave Motion. 11, 23 − 32, (1989). https://doi.org/10.1016/0165-2125(89)90010-3
Lasiecka, I., Messaoudi, S. A., Mustafa, M. I., Note on intrinsic decay rates for abstract wave equations with memory. J. Math. Phys. 54, 031504, (2013). https://doi.org/10.1063/1.4793988
Lebon, G., Cloot, A., Propagation of ultrasonic sound waves in dissipative dilute gases and extended irreversible thermodynamics. Wave Motion 11,23 − 32, (1989). https://doi.org/10.1016/0165-2125(89)90010-3
Medjden, M., Tatar, N.: Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel. Applied Mathematics and Computation 167, 1221 − 1235, (2005). https://doi.org/10.1016/j.amc.2004.08.035
Mesloub, F., Boulaaras, S.: General decay for a viscoelastic problem with not necessarily decreasing kernel. J. Appl. Math. Comput. 58, 647-665, (2018). https://doi.org/10.1007/s12190-017-1161-9
Marchand, R., McDevitt, T., Triggiani, R., 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(15), 1896 − 1929, (2012). https://doi.org/10.1002/mma.1576
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