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doi: 10.3934/eect.2021038
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Existence and general decay of Balakrishnan-Taylor viscoelastic equation with nonlinear frictional damping and logarithmic source term

1. 

The Preparatory Year Program, Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran, KSA

2. 

Department of Mathematics, Faculty of Science and Arts, King Khalid University, Mohail Assir, KSA

3. 

Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

4. 

Department of Mathematics, University of Gabès, Gabès

5. 

Department of Mathematics, College of Sciences, University of Sharjah, P.O.Box 27272, Sharjah, UAE

Received  September 2020 Revised  May 2021 Early access August 2021

Fund Project: The third author is supported by the National Natural Science Foundation of China (No. 11701465) and the fifth author is supported by Research Group MASEP

In this paper, we consider a Balakrishnan-Taylor viscoelastic wave equation with nonlinear frictional damping and logarithmic source term. By assuming a more general type of relaxation functions, we establish explicit and general decay rate results, using the multiplier method and some properties of the convex functions. This result is new and generalizes earlier results in the literature.

Citation: Mohammad Al-Gharabli, Mohamed Balegh, Baowei Feng, Zayd Hajjej, Salim A. Messaoudi. Existence and general decay of Balakrishnan-Taylor viscoelastic equation with nonlinear frictional damping and logarithmic source term. Evolution Equations and Control Theory, doi: 10.3934/eect.2021038
References:
[1]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Math. Acad. Sci. Paris, 347 (2009), 867-872.  doi: 10.1016/j.crma.2009.05.011.

[2]

M. M. Al-GharabliA. Guesmia and S. A. Messaoudi, Well-posedness and asymptotic stability results for a viscoelastic plate equation with a logarithmic nonlinearity, Applicable Analysis, 99 (2020), 50-74.  doi: 10.1080/00036811.2018.1484910.

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Ranslated from the Russian by K. Vogtmann and A. Weinstein. Second Edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[4]

A. V. Balakrishnan and L. W. Taylor, Distributed Parameter Nonlinear Damping Models for Flight Structures, In: Proceedings ``Daming 89", Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989.

[5]

K. Bartkowski and P. Gorka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A., 41 (2008), 11 pp. doi: 10.1088/1751-8113/41/35/355201.

[6]

R. W. Bass and D. Zes, Spillover, nonlinearity and flexible structures, Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, 2 (1991), 1633-1637.  doi: 10.1109/CDC.1991.261683.

[7]

I. Bialynicki-Birula and J. Mycielsk, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. Ser Sci. Math. Astronom. Phys., 23 (1975), 461-466. 

[8]

I. Bialynicki-Birula and J. Mycielsk, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.

[9]

M. M. CavalcantiV. N. D. CavalcantiI. Lasiecka and F. A. Nascimento, Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects, Discrete Conti. Dyn. Syst. Ser. B., 19 (2014), 1987-2012.  doi: 10.3934/dcdsb.2014.19.1987.

[10]

M. M. CavalcantiA. D. D. CavalcantiI. Lasiecka and X. Wang, Existence and sharp decay rate estimates for a von Karman system with long memory, Nonlinear Anal.: Real World Appl., 22 (2015), 289-306.  doi: 10.1016/j.nonrwa.2014.09.016.

[11]

M. M. CavalcantiV. N. D. Cavalcanti and J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Methods Appl. Sci., 24 (2001), 1043-1053.  doi: 10.1002/mma.250.

[12]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.  doi: 10.1137/S0363012902408010.

[13]

T. Cazenave and A. Haraux, Equations d'evolution avec non-linearite logarithmique, Ann. Fac. Sci. Toulouse Math., 5 (1980), 21-51.  doi: 10.5802/afst.543.

[14]

H. R. Clark, Elastic membrane equation in bounded and unbounded domains, Electron J. Qual. Theory Differ. Equ., 11 (2002), 1-21. 

[15]

B. Feng and Y. H. Kang, Decay rates for a viscoelastic wave equation with Balakrishnan-Taylor and frictional dampings, Topol. Methods Nonlinear Anal., 54 (2019), 321-343.  doi: 10.12775/tmna.2019.047.

[16]

B. Feng and A. Soufyane, Existence and decay rates for a coupled Balakrishnan-Taylor viscoelastic system with dynamic boundary conditions, Math. Methods Appl. Sci., 43 (2020), 3375-3391.  doi: 10.1002/mma.6127.

[17]

P. Gorka, Logarithmic Klein-Gordon equation, Acta Phys. Polon. B., 40 (2009), 59-66. 

[18]

L. Gross, Logarithmic Sobolev inequalities, Am. J. Math., 97 (1975), 1061-1083.  doi: 10.2307/2373688.

[19]

T. G. Ha, Stabilization for the wave equation with variable coefficients and Balakrishnan-Taylor damping, Taiwan. J. Math., 21 (2017), 807-817.  doi: 10.11650/tjm/7828.

[20]

T. G. Ha, Asymptotic stability of the viscoelastic equation with variable coefficients and the Balakrishnan-Taylor damping, Taiwan. J. Math., 22 (2018), 931-948.  doi: 10.11650/tjm/171203.

[21]

T. G. Ha, General decay rate estimates for viscoelastic wave equation with Balakrishnan-Taylor damping, Z. Angew. Math. Phys., 67 (2016), 17 pp. doi: 10.1007/s00033-016-0625-3.

[22]

X. Han, Global existence of weak solutions for a logarithmic wave equation arising from Q-ball dynamics, Bull. Korean Math. Soc., 50 (2013), 275-283.  doi: 10.4134/BKMS.2013.50.1.275.

[23]

T. Hiramatsu, M. Kawasaki and F. Takahashi, Numerical study of Q-ball formation in gravity mediation, J. Cosmol. Astropart. Phys., 2010 (2010).

[24]

Q. HuH. Zhang and G. Liu, Asymptotic behavior for a class of logarithmic wave equations with linear damping, Appl. Math. Optim., 79 (2019), 131-144.  doi: 10.1007/s00245-017-9423-3.

[25]

V. Komornik, Exact controllability and stabilisation, the multiplier method, RMA, Masson, Paris, 36 (1994).

[26]

I. Lasiecka, S. A. Messaoudi and M. I. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory, J. Math. Phys., 54 (2013), 031504. doi: 10.1063/1.4793988.

[27]

I. Lasiecka and X. Wang, Intrinsic decay rate estimates for semilinear abstract second order equations with memory, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer INdAM Series, Cham: Springer, 10 (2014), 271-303.  doi: 10.1007/978-3-319-11406-4_14.

[28]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533. 

[29]

W. J. Liu and E. Zuazua, Deacy rates for dissipative wave equations, Papers in Memory of Ennio De Giorgi (Italian). Ricerche Mat., 48 (1999), 61-75. 

[30]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var., 4 (1999), 419-444.  doi: 10.1051/cocv:1999116.

[31]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut., 12 (1999), 251-283.  doi: 10.5209/rev_REMA.1999.v12.n1.17227.

[32]

S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467.  doi: 10.1016/j.jmaa.2007.11.048.

[33]

S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal., 69 (2008), 2589-2598.  doi: 10.1016/j.na.2007.08.035.

[34]

M. I. Mustafa, On the control of the wave equation by memory-type boundary condition, Discrete Contin. Dyn. Syst. Ser. A., 35 (2015), 1179-1192.  doi: 10.3934/dcds.2015.35.1179.

[35]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Methods Appl. Sci., 41 (2018), 192-204.  doi: 10.1002/mma.4604.

[36]

M. I. Mustafa, General decay result for nonlinear viscoelastic equations, J. Math. Anal. Appl., 457 (2018), 134-152.  doi: 10.1016/j.jmaa.2017.08.019.

[37]

M. I. Mustafa and G. A. Abusharkh, Plate equations with frictional and viscoelastic dampings, Appl. Anal., 96 (2017), 1170-1187.  doi: 10.1080/00036811.2016.1178724.

[38]

M. I. Mustafa and S. A. Messaoudi, General energy decay rates for a weakly damped wave equation, Commun. Math. Anal., 9 (2010), 67-76. 

[39]

M. I. Mustafa and S. A. Messaoudi, General stability result for viscoelastic wave equations, J. Math. Phys., 53 (2012), 053702. doi: 10.1063/1.4711830.

[40]

S. H. Park, Arbitrary decay of energy for a viscoelastic problem with Balakrishnan-Taylor damping, Taiwan. J. Math., 20 (2016), 129-141.  doi: 10.11650/tjm.20.2016.6079.

[41]

A. Peyravi, General stability and exponential growth for a class of semi-linear wave equations with logarithmic source and memory terms, Appl. Math. Optim., 81 (2020), 545-561.  doi: 10.1007/s00245-018-9508-7.

[42]

N-e. Tatar and A. Zaraï, Exponential stability and blow up for a problem with Balakrishnan-Taylor damping, Demonstratio Math., 44 (2011), 67-90. 

[43]

N-e. Tatar and A. Zaraï, On a Kirchhoff equation with Balakrishnan-Taylor damping and source term, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 18 (2011), 615-627. 

[44]

N-e. Tatar and A. Zaraï, Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping, Arch. Math. (Brno), 46 (2010), 47-56. 

[45]

S. T. Wu, General decay of solutions for a viscoelastic equation with Balakrishnan-Taylor damping and nonlinear boundary damping-source interactions, Acta Math. Sci. Ser. B (Engl. Ed.), 35 (2015), 981-994. 

[46]

T. J. Xiao and J. Liang, Coupled second order semilinear evolution equations indirectly damped via memory effects, J. Differ. Equ., 254 (2013), 2128-2157.  doi: 10.1016/j.jde.2012.11.019.

[47]

Y. You, Intertial manifolds and stabilization of nonlinear beam equations with Balakrishnan-Taylor damping, Abstr. Appl. Anal., 1 (1996), 83-102.  doi: 10.1155/S1085337596000048.

[48]

A. Zaraï and N. Tatar, Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping, Arch. Math. (Brno), 46 (2010), 157-176. 

[49]

A. Zaraï and N. Tatar, Non-solvability of Balakrishnan-Taylor equation with memory term in $\mathbb{R}^N$, Advances in Applied Mathematics and Approximation Theory, Springer Proc. Math. Stat., Springer, New York, 41 (2013), 411-419.  doi: 10.1007/978-1-4614-6393-1_27.

show all references

References:
[1]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Math. Acad. Sci. Paris, 347 (2009), 867-872.  doi: 10.1016/j.crma.2009.05.011.

[2]

M. M. Al-GharabliA. Guesmia and S. A. Messaoudi, Well-posedness and asymptotic stability results for a viscoelastic plate equation with a logarithmic nonlinearity, Applicable Analysis, 99 (2020), 50-74.  doi: 10.1080/00036811.2018.1484910.

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Ranslated from the Russian by K. Vogtmann and A. Weinstein. Second Edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[4]

A. V. Balakrishnan and L. W. Taylor, Distributed Parameter Nonlinear Damping Models for Flight Structures, In: Proceedings ``Daming 89", Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989.

[5]

K. Bartkowski and P. Gorka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A., 41 (2008), 11 pp. doi: 10.1088/1751-8113/41/35/355201.

[6]

R. W. Bass and D. Zes, Spillover, nonlinearity and flexible structures, Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, 2 (1991), 1633-1637.  doi: 10.1109/CDC.1991.261683.

[7]

I. Bialynicki-Birula and J. Mycielsk, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. Ser Sci. Math. Astronom. Phys., 23 (1975), 461-466. 

[8]

I. Bialynicki-Birula and J. Mycielsk, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.

[9]

M. M. CavalcantiV. N. D. CavalcantiI. Lasiecka and F. A. Nascimento, Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects, Discrete Conti. Dyn. Syst. Ser. B., 19 (2014), 1987-2012.  doi: 10.3934/dcdsb.2014.19.1987.

[10]

M. M. CavalcantiA. D. D. CavalcantiI. Lasiecka and X. Wang, Existence and sharp decay rate estimates for a von Karman system with long memory, Nonlinear Anal.: Real World Appl., 22 (2015), 289-306.  doi: 10.1016/j.nonrwa.2014.09.016.

[11]

M. M. CavalcantiV. N. D. Cavalcanti and J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Methods Appl. Sci., 24 (2001), 1043-1053.  doi: 10.1002/mma.250.

[12]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.  doi: 10.1137/S0363012902408010.

[13]

T. Cazenave and A. Haraux, Equations d'evolution avec non-linearite logarithmique, Ann. Fac. Sci. Toulouse Math., 5 (1980), 21-51.  doi: 10.5802/afst.543.

[14]

H. R. Clark, Elastic membrane equation in bounded and unbounded domains, Electron J. Qual. Theory Differ. Equ., 11 (2002), 1-21. 

[15]

B. Feng and Y. H. Kang, Decay rates for a viscoelastic wave equation with Balakrishnan-Taylor and frictional dampings, Topol. Methods Nonlinear Anal., 54 (2019), 321-343.  doi: 10.12775/tmna.2019.047.

[16]

B. Feng and A. Soufyane, Existence and decay rates for a coupled Balakrishnan-Taylor viscoelastic system with dynamic boundary conditions, Math. Methods Appl. Sci., 43 (2020), 3375-3391.  doi: 10.1002/mma.6127.

[17]

P. Gorka, Logarithmic Klein-Gordon equation, Acta Phys. Polon. B., 40 (2009), 59-66. 

[18]

L. Gross, Logarithmic Sobolev inequalities, Am. J. Math., 97 (1975), 1061-1083.  doi: 10.2307/2373688.

[19]

T. G. Ha, Stabilization for the wave equation with variable coefficients and Balakrishnan-Taylor damping, Taiwan. J. Math., 21 (2017), 807-817.  doi: 10.11650/tjm/7828.

[20]

T. G. Ha, Asymptotic stability of the viscoelastic equation with variable coefficients and the Balakrishnan-Taylor damping, Taiwan. J. Math., 22 (2018), 931-948.  doi: 10.11650/tjm/171203.

[21]

T. G. Ha, General decay rate estimates for viscoelastic wave equation with Balakrishnan-Taylor damping, Z. Angew. Math. Phys., 67 (2016), 17 pp. doi: 10.1007/s00033-016-0625-3.

[22]

X. Han, Global existence of weak solutions for a logarithmic wave equation arising from Q-ball dynamics, Bull. Korean Math. Soc., 50 (2013), 275-283.  doi: 10.4134/BKMS.2013.50.1.275.

[23]

T. Hiramatsu, M. Kawasaki and F. Takahashi, Numerical study of Q-ball formation in gravity mediation, J. Cosmol. Astropart. Phys., 2010 (2010).

[24]

Q. HuH. Zhang and G. Liu, Asymptotic behavior for a class of logarithmic wave equations with linear damping, Appl. Math. Optim., 79 (2019), 131-144.  doi: 10.1007/s00245-017-9423-3.

[25]

V. Komornik, Exact controllability and stabilisation, the multiplier method, RMA, Masson, Paris, 36 (1994).

[26]

I. Lasiecka, S. A. Messaoudi and M. I. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory, J. Math. Phys., 54 (2013), 031504. doi: 10.1063/1.4793988.

[27]

I. Lasiecka and X. Wang, Intrinsic decay rate estimates for semilinear abstract second order equations with memory, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer INdAM Series, Cham: Springer, 10 (2014), 271-303.  doi: 10.1007/978-3-319-11406-4_14.

[28]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533. 

[29]

W. J. Liu and E. Zuazua, Deacy rates for dissipative wave equations, Papers in Memory of Ennio De Giorgi (Italian). Ricerche Mat., 48 (1999), 61-75. 

[30]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var., 4 (1999), 419-444.  doi: 10.1051/cocv:1999116.

[31]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut., 12 (1999), 251-283.  doi: 10.5209/rev_REMA.1999.v12.n1.17227.

[32]

S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467.  doi: 10.1016/j.jmaa.2007.11.048.

[33]

S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal., 69 (2008), 2589-2598.  doi: 10.1016/j.na.2007.08.035.

[34]

M. I. Mustafa, On the control of the wave equation by memory-type boundary condition, Discrete Contin. Dyn. Syst. Ser. A., 35 (2015), 1179-1192.  doi: 10.3934/dcds.2015.35.1179.

[35]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Methods Appl. Sci., 41 (2018), 192-204.  doi: 10.1002/mma.4604.

[36]

M. I. Mustafa, General decay result for nonlinear viscoelastic equations, J. Math. Anal. Appl., 457 (2018), 134-152.  doi: 10.1016/j.jmaa.2017.08.019.

[37]

M. I. Mustafa and G. A. Abusharkh, Plate equations with frictional and viscoelastic dampings, Appl. Anal., 96 (2017), 1170-1187.  doi: 10.1080/00036811.2016.1178724.

[38]

M. I. Mustafa and S. A. Messaoudi, General energy decay rates for a weakly damped wave equation, Commun. Math. Anal., 9 (2010), 67-76. 

[39]

M. I. Mustafa and S. A. Messaoudi, General stability result for viscoelastic wave equations, J. Math. Phys., 53 (2012), 053702. doi: 10.1063/1.4711830.

[40]

S. H. Park, Arbitrary decay of energy for a viscoelastic problem with Balakrishnan-Taylor damping, Taiwan. J. Math., 20 (2016), 129-141.  doi: 10.11650/tjm.20.2016.6079.

[41]

A. Peyravi, General stability and exponential growth for a class of semi-linear wave equations with logarithmic source and memory terms, Appl. Math. Optim., 81 (2020), 545-561.  doi: 10.1007/s00245-018-9508-7.

[42]

N-e. Tatar and A. Zaraï, Exponential stability and blow up for a problem with Balakrishnan-Taylor damping, Demonstratio Math., 44 (2011), 67-90. 

[43]

N-e. Tatar and A. Zaraï, On a Kirchhoff equation with Balakrishnan-Taylor damping and source term, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 18 (2011), 615-627. 

[44]

N-e. Tatar and A. Zaraï, Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping, Arch. Math. (Brno), 46 (2010), 47-56. 

[45]

S. T. Wu, General decay of solutions for a viscoelastic equation with Balakrishnan-Taylor damping and nonlinear boundary damping-source interactions, Acta Math. Sci. Ser. B (Engl. Ed.), 35 (2015), 981-994. 

[46]

T. J. Xiao and J. Liang, Coupled second order semilinear evolution equations indirectly damped via memory effects, J. Differ. Equ., 254 (2013), 2128-2157.  doi: 10.1016/j.jde.2012.11.019.

[47]

Y. You, Intertial manifolds and stabilization of nonlinear beam equations with Balakrishnan-Taylor damping, Abstr. Appl. Anal., 1 (1996), 83-102.  doi: 10.1155/S1085337596000048.

[48]

A. Zaraï and N. Tatar, Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping, Arch. Math. (Brno), 46 (2010), 157-176. 

[49]

A. Zaraï and N. Tatar, Non-solvability of Balakrishnan-Taylor equation with memory term in $\mathbb{R}^N$, Advances in Applied Mathematics and Approximation Theory, Springer Proc. Math. Stat., Springer, New York, 41 (2013), 411-419.  doi: 10.1007/978-1-4614-6393-1_27.

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