January  2019, 18(1): 159-180. doi: 10.3934/cpaa.2019009

Existence and a general decay results for a viscoelastic plate equation with a logarithmic nonlinearity

1. 

King Fahd University of Petroleum and Minerals, The Preparatory Year Program, Department of Mathematics, Dhahran 31261, Saudi Arabia

2. 

Institut Elie Cartan de Lorraine, UMR 7502, Université de Lorraine, 3 Rue Augustin Fresnel, BP 45112, 57073 Metz Cedex 03, France

3. 

King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran 31261, Saudi Arabia

Received  September 2017 Revised  January 2018 Published  August 2018

In this paper, we consider a viscoelastic plate equation with a logarithmic nonlinearity. Using the Galaerkin method and the multiplier method, we establish the existence of solutions and prove an explicit and general decay rate result. This result extends and improves many results in the literature such as Gorka [19], Hiramatsu et al. [27] and Han and Wang [26].

Citation: Mohammad M. Al-Gharabli, Aissa Guesmia, Salim A. Messaoudi. Existence and a general decay results for a viscoelastic plate equation with a logarithmic nonlinearity. Communications on Pure and Applied Analysis, 2019, 18 (1) : 159-180. doi: 10.3934/cpaa.2019009
References:
[1]

J. Barrow and P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D, 52 (1995), 5576-5587. 

[2]

K. Bartkowski and P. Gorka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities J. Phys. A, 41 (2008), 355201, 11 pp. doi: 10. 1088/1751-8113/41/35/355201.

[3]

A. Benaissa and A. Guesmia, Energy decay of solutions of a wave equation of ϕ-Laplacian type with a general weakly nolinear dissipation, Elec. J. Diff. Equa., 109 (2008), 1-22. 

[4]

S. Berrimi and S. Messaoudi, Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping, Electron. J. Differential Equations, 88 (2004), 1-10. 

[5]

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

[6]

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

[7]

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

[8]

M. CavalcantiV. Domingos Cavalcanti and J. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, E. J. Differ. Eq., 44 (2002), 1-14. 

[9]

M. Cavalcanti and A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type, Diff. Integ. Equa., 18 (2005), 583-600. 

[10]

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

[11]

T. Cazenave and A. Haraux, Equations d'evolution avec non-linearite logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21-51. 

[12]

H. ChenP. Luo and G. W. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98.  doi: 10.1016/j.jmaa.2014.08.030.

[13]

W. Chen and Y. Zhou, Global nonexistence for a semilinear Petrovsky equation, Nonlinear Analysis A, 70 (2009), 3203-3208.  doi: 10.1016/j.na.2008.04.024.

[14]

R. Christensen, Theory of Viscoelasticity, An Introduction, Academic Press: New York, 1982.

[15]

C. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[16]

C. Dafermos, On abstract volterra equations with applications to linear viscoelasticity, J. Differ. Equ., 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4.

[17]

G. Dasios and F. Zafiropoulos, Equipartition of energy in linearized 3-D viscoelasticity, Quart. Appl. Math., 48 (1990), 715-730.  doi: 10.1090/qam/1079915.

[18]

K. Enqvist and J. McDonald, Q-balls and baryogenesis in the MSSM, Phys. Lett. B, 425 (1998), 309-321. 

[19]

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

[20]

P. GorkaH. Prado and G. Reyes, Nonlinear equations with infinitely many derivatives, Complex Anal. Oper. Theory, 5 (2011), 313-323.  doi: 10.1007/s11785-009-0043-z.

[21]

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

[22]

A. Guesmia, Existence globale et stabilisation interne non linéaire d'un système de Petrovsky, Bull. Belg. Math. Soc., 5 (1998), 583-594. 

[23]

A. Guesmia, Stabilisation de l'équation des ondes avec conditions aux limites de type mémoire, Afrika Matematika, 10 (1999), 14-25. 

[24]

A. GuesmiaS. Messaoudi and B. Said-Houari, General decay of solutions of a nonlinear system of viscoelastic wave equations, NoDEA, 18 (2011), 659-684.  doi: 10.1007/s00030-011-0112-7.

[25]

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.

[26]

X. Han and M. Wang, General decay estimate of energy for the second order evolution equations with memory, Act Appl. Math., 110 (2010), 194-207.  doi: 10.1007/s10440-008-9397-x.

[27]

T. Hiramatsu, M. Kawasaki and F. Takahashi, Numerical study of Q-ball formation in gravity mediation, Journal of Cosmology and Astroparticle Physics, 6 (2010), 008.

[28]

H. Hrusa, Global existence and asymptotic stability for a semilinear Volterra equation with large initial data, SIAM J. Math. Anal., 16 (1985), 110-134.  doi: 10.1137/0516007.

[29]

V. Komornik, On the nonlinear boundary stabilization of Kirchoff plates, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 323-337.  doi: 10.1007/BF01194984.

[30]

J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, 1989. doi: 10. 1137/1. 9781611970821.

[31]

J. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, International Series of Numerical Mathematics, vol. 91. Birhauser: Verlag, Bassel, 1989.

[32]

I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli moments only, J. Differential Equations, 95 (1992), 169-182.  doi: 10.1016/0022-0396(92)90048-R.

[33]

J. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, second Edition, Dunod, Paris, 2002.

[34]

Z. LiZ. Zhao and Y. Chen, Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation, Nonlinear Anal.: RealWorld Applications, 12 (2011), 1759-1773.  doi: 10.1016/j.nonrwa.2010.11.009.

[35]

M-T. Lacroix-Sonrier, Distrubutions Espace de Sobolev Application, Ellipses Edition Marketing S. A, 1998.

[36]

S. Messaoudi, Global existence and nonexistence in a system of Petrovsky, Journal of Mathematical Analysis and Applications, 265 (2002), 296-308.  doi: 10.1006/jmaa.2001.7697.

[37]

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

[38]

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

[39]

S. Messaoudi and N.-E Tatar, Global existence asymptotic behavior for a non-linear viscoelastic problem, Math. Methods Sci. Res., 7 (2003), 136-149. 

[40]

S. Messaoudi and N.-E Tatar, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Math. Methods Appl. Sci., 30 (2007), 665-680.  doi: 10.1002/mma.804.

[41]

S. Messaoudi and W. Al-Khulaifi, General and optimal decay for a quasilinear viscoelastic equation, Applied Mathematics Letters, 66 (2017), 16-22.  doi: 10.1016/j.aml.2016.11.002.

[42]

Rivera J. Muñoz, Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648.  doi: 10.1090/qam/1306041.

[43]

Rivera J. MuñozE. C. Lapa and R. Barreto, Decay rates for viscoelastic paltes with memory, Journal of Elasticity, 44 (1996), 61-87.  doi: 10.1007/BF00042192.

[44]

M. Santos and F. junior, A boundary condition with memory for Kirchoff plates equations, Appl. Math. Comput., 148 (2004), 475-496.  doi: 10.1016/S0096-3003(02)00915-3.

[45]

V. S. Vladimirov, The equation of the p-adic open string for the scalar tachyon field, Izv. Math., 69 (2005), 487-512.  doi: 10.1070/IM2005v069n03ABEH000536.

show all references

References:
[1]

J. Barrow and P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D, 52 (1995), 5576-5587. 

[2]

K. Bartkowski and P. Gorka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities J. Phys. A, 41 (2008), 355201, 11 pp. doi: 10. 1088/1751-8113/41/35/355201.

[3]

A. Benaissa and A. Guesmia, Energy decay of solutions of a wave equation of ϕ-Laplacian type with a general weakly nolinear dissipation, Elec. J. Diff. Equa., 109 (2008), 1-22. 

[4]

S. Berrimi and S. Messaoudi, Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping, Electron. J. Differential Equations, 88 (2004), 1-10. 

[5]

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

[6]

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

[7]

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

[8]

M. CavalcantiV. Domingos Cavalcanti and J. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, E. J. Differ. Eq., 44 (2002), 1-14. 

[9]

M. Cavalcanti and A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type, Diff. Integ. Equa., 18 (2005), 583-600. 

[10]

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

[11]

T. Cazenave and A. Haraux, Equations d'evolution avec non-linearite logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21-51. 

[12]

H. ChenP. Luo and G. W. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98.  doi: 10.1016/j.jmaa.2014.08.030.

[13]

W. Chen and Y. Zhou, Global nonexistence for a semilinear Petrovsky equation, Nonlinear Analysis A, 70 (2009), 3203-3208.  doi: 10.1016/j.na.2008.04.024.

[14]

R. Christensen, Theory of Viscoelasticity, An Introduction, Academic Press: New York, 1982.

[15]

C. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[16]

C. Dafermos, On abstract volterra equations with applications to linear viscoelasticity, J. Differ. Equ., 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4.

[17]

G. Dasios and F. Zafiropoulos, Equipartition of energy in linearized 3-D viscoelasticity, Quart. Appl. Math., 48 (1990), 715-730.  doi: 10.1090/qam/1079915.

[18]

K. Enqvist and J. McDonald, Q-balls and baryogenesis in the MSSM, Phys. Lett. B, 425 (1998), 309-321. 

[19]

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

[20]

P. GorkaH. Prado and G. Reyes, Nonlinear equations with infinitely many derivatives, Complex Anal. Oper. Theory, 5 (2011), 313-323.  doi: 10.1007/s11785-009-0043-z.

[21]

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

[22]

A. Guesmia, Existence globale et stabilisation interne non linéaire d'un système de Petrovsky, Bull. Belg. Math. Soc., 5 (1998), 583-594. 

[23]

A. Guesmia, Stabilisation de l'équation des ondes avec conditions aux limites de type mémoire, Afrika Matematika, 10 (1999), 14-25. 

[24]

A. GuesmiaS. Messaoudi and B. Said-Houari, General decay of solutions of a nonlinear system of viscoelastic wave equations, NoDEA, 18 (2011), 659-684.  doi: 10.1007/s00030-011-0112-7.

[25]

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.

[26]

X. Han and M. Wang, General decay estimate of energy for the second order evolution equations with memory, Act Appl. Math., 110 (2010), 194-207.  doi: 10.1007/s10440-008-9397-x.

[27]

T. Hiramatsu, M. Kawasaki and F. Takahashi, Numerical study of Q-ball formation in gravity mediation, Journal of Cosmology and Astroparticle Physics, 6 (2010), 008.

[28]

H. Hrusa, Global existence and asymptotic stability for a semilinear Volterra equation with large initial data, SIAM J. Math. Anal., 16 (1985), 110-134.  doi: 10.1137/0516007.

[29]

V. Komornik, On the nonlinear boundary stabilization of Kirchoff plates, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 323-337.  doi: 10.1007/BF01194984.

[30]

J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, 1989. doi: 10. 1137/1. 9781611970821.

[31]

J. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, International Series of Numerical Mathematics, vol. 91. Birhauser: Verlag, Bassel, 1989.

[32]

I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli moments only, J. Differential Equations, 95 (1992), 169-182.  doi: 10.1016/0022-0396(92)90048-R.

[33]

J. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, second Edition, Dunod, Paris, 2002.

[34]

Z. LiZ. Zhao and Y. Chen, Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation, Nonlinear Anal.: RealWorld Applications, 12 (2011), 1759-1773.  doi: 10.1016/j.nonrwa.2010.11.009.

[35]

M-T. Lacroix-Sonrier, Distrubutions Espace de Sobolev Application, Ellipses Edition Marketing S. A, 1998.

[36]

S. Messaoudi, Global existence and nonexistence in a system of Petrovsky, Journal of Mathematical Analysis and Applications, 265 (2002), 296-308.  doi: 10.1006/jmaa.2001.7697.

[37]

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

[38]

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

[39]

S. Messaoudi and N.-E Tatar, Global existence asymptotic behavior for a non-linear viscoelastic problem, Math. Methods Sci. Res., 7 (2003), 136-149. 

[40]

S. Messaoudi and N.-E Tatar, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Math. Methods Appl. Sci., 30 (2007), 665-680.  doi: 10.1002/mma.804.

[41]

S. Messaoudi and W. Al-Khulaifi, General and optimal decay for a quasilinear viscoelastic equation, Applied Mathematics Letters, 66 (2017), 16-22.  doi: 10.1016/j.aml.2016.11.002.

[42]

Rivera J. Muñoz, Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648.  doi: 10.1090/qam/1306041.

[43]

Rivera J. MuñozE. C. Lapa and R. Barreto, Decay rates for viscoelastic paltes with memory, Journal of Elasticity, 44 (1996), 61-87.  doi: 10.1007/BF00042192.

[44]

M. Santos and F. junior, A boundary condition with memory for Kirchoff plates equations, Appl. Math. Comput., 148 (2004), 475-496.  doi: 10.1016/S0096-3003(02)00915-3.

[45]

V. S. Vladimirov, The equation of the p-adic open string for the scalar tachyon field, Izv. Math., 69 (2005), 487-512.  doi: 10.1070/IM2005v069n03ABEH000536.

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