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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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