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January  2020, 19(1): 455-492. doi: 10.3934/cpaa.2020023

Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions

Nguyen Thanh Long 1,, , Hoang Hai Ha 2, , Le Thi Phuong Ngoc 3, and  Nguyen Anh Triet 4,

1. 

Department of Mathematics and Computer Science, VNUHCM-University of Science, 227 Nguyen Van Cu Str., Dist.5, Ho Chi Minh City, Vietnam
2. 

Faculty of Applied Science, Ho Chi Minh City University of Technology, Vietnam National University Ho Chi Minh City, 268 Ly Thuong Kiet Str., Dist. 10, Ho Chi Minh City, Vietnam
3. 

University of Khanh Hoa, 01 Nguyen Chanh Str., Nha Trang City, Vietnam
4. 

Department of Mathematics, University of Architecture of Ho Chi Minh City, 196 Pasteur Str., Dist. 3, Ho Chi Minh City, Vietnam

* Corresponding author

Received  January 2019 Revised  May 2019 Published  July 2019

This paper is devoted to the study of a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Based on the Faedo-Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we first establish two local existence theorems of weak solutions. By the construction of a suitable Lyapunov functional, we next prove a blow up result and a decay result of global solutions.

Keywords: System of nonlinear viscoelastic equations, nonlinear boundary conditions, Faedo-Galerkin method, blow-up and exponential decay estimates.
Mathematics Subject Classification: Primary: 35L05, 35L15, 35L20, 35L55, 35L70.
Citation: Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023
References:
[1]

M. Bergounioux, N. T. Long and Alain P. N. Dinh, Mathematical model for a shock problem involving a linear viscoelastic bar, Nonlinear Anal., 43 (2001), 547–561. doi: 10.1016/S0362-546X(99)00218-7.

[2]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho and J. A. Soriano, Existence and uniform decay of solutions of a degenerate equation with nonlinear boundary damping and boundary memory source term, Nonlinear Anal., 38 (1999), 281–294. doi: 10.1016/S0362-546X(98)00195-3.

[3]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, On the existence and the uniform decay of a hyperbolic, Southeast Asian Bulletin of Mathematics, 24 (2000), 183–199. doi: 10.1007/s100120070002.

[4]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and M. L. Santos, Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary, Applied Mathematics and Computation, 150 (2004), 439–465. doi: 10.1016/S0096-3003(03)00284-4.

[5]

Fei Liang and Hongjun Gao, Global nonexistence of positive initial-energy solutions for coupled nonlinear wave equations with damping and source terms, Abstract and Applied Analysis, Vol. 2011, Art. ID 760209, 14 pages. doi: 10.1155/2011/760209.

[6]

V. A. Khoa, L. T. P. Ngoc and N. T. Long, Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms, Evolution Equations & Control Theory, 8 (2019), 359–395.

[7]

V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Vol. 1. Academic Press, NewYork, 1969.

[8]

C. Sideris and Yi Zhou, Almost global existence for 2-D incompressible isotropic elastodynamics, Trans. Amer. Math. Soc., 367 (2015), 8175–8197. doi: 10.1090/tran/6294.

[9]

Zhen Lei, On 2D Viscoelasticity with Small Strain, Arch. Rational Mech. Anal., 198 (2010), 13-37. doi: 10.1007/s00205-010-0346-2.

[10]

J. L. Lions, Quelques méthodes de ré solution des problèmes aux limites nonlinéaires, Dunod; Gauthier-Villars, Paris, 1969.

[11]

W. Liu, G. Li and L. Hong, General decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping, Journal of Function Spaces, Vol. 2014, Art. ID 284809, 21 pages. doi: 10.1155/2014/284809.

[12]

N. T. Long and Alain P. N. Dinh, On the quasilinear wave equation: $u_tt-\Delta u+f(u, $ $u_{t}) = 0$ associated with a mixed nonhomogeneous condition, Nonlinear Anal., 19 (1992). doi: 10.1016/0362-546X(92)90097-X.

[13]

N. T. Long and Alain P. N. Dinh, A semilinear wave equation associated with a linear differential equation with Cauchy data, Nonlinear Anal. TMA., 24 (1995), 1261–1279. doi: 10.1016/0362-546X(94)00196-O.

[14]

N. T. Long and T. N. Diem, On the nonlinear wave equation $u_tt-u_xx = f(x, $ $t, \, u, $ $u_{x}, $ $u_{t})$ associated with the mixed homogeneous conditions, Nonlinear Anal. TMA., 29 (1997), 1217–1230. doi: 10.1016/S0362-546X(97)87360-9.

[15]

N. T. Long, Alain P. N. Dinh and T. N. Diem, On a shock problem involving a nonlinear viscoelastic bar, J. Boundary Value Problems, 2005 (2005), 337–358. doi: 10.1155/bvp.2005.337.

[16]

N. T. Long and L. T. P. Ngoc, On a nonlinear wave equation with boundary conditions of two-point type, J. Math. Anal. Appl., 385 (2012), 1070–1093. doi: 10.1016/j.jmaa.2011.07.034.

[17]

S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260 (2003), 58–66. doi: 10.1002/mana.200310104.

[18]

Changxing Miao and Youbin Zhu, Global smooth solutions for a nonlinear system of wave equations, Nonlinear Anal., 67 (2007), 3136-3151. doi: 10.1016/j.na.2006.10.006.

[19]

L. T. P. Ngoc and N. T. Long, Existence, blow-up and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions, Mathematical Methods in the Applied Sciences, 37 (2014), 464–487. doi: 10.1002/mma.2803.

[20]

L. T. P. Ngoc and N. T. Long, Existence and exponential decay for a nonlinear wave equation with a nonlocal boundary condition, Communications on Pure and Applied Analysis, 12 (2013), 2001–2029. doi: 10.3934/cpaa.2013.12.2001.

[21]

L. T. P. Ngoc, C. H. Hoa and N. T. Long, Existence, blow-up, and exponential decay estimates for a system of semilinear wave equations associated with the helical flows of Maxwell fluid, Mathematical Methods in the Applied Sciences, 39 (2016), 2334–2357. doi: 10.1002/mma.3643.

[22]

L. T. P. Ngoc, N. A. Triet, Alain P. N. Dinh and N. T. Long, Existence and exponential decay of solutions for a wave equation with integral nonlocal boundary conditions of memory type, Numerical Functional Analysis and Optimization, 38 (2017), 1173–1207. doi: 10.1080/01630563.2017.1320672.

[23]

L. T. P. Ngoc, L. N. K. Hang and N. T. Long, On a nonlinear wave equation associated with the boundary conditions involving convolution, Nonlinear Anal. TMA., 70 (2009), 3943–3965. doi: 10.1016/j.na.2008.08.004.

[24]

M. L. Santos, Decay rates for solutions of a system of wave equations with memory, Electronic J. Differential Equations, 38 (2002), 1–17.

[25]

C. Sideris, Global behavior of solutions to nonlinear wave equations in three dimensions, Comm. P.D.E., 8 (1983), 1291–1323. doi: 10.1080/03605308308820304.

[26]

C. Sideris, The null condition and global existence of nonlinear elastic waves, Invent. Math., 123 (1996), 323–342. doi: 10.1007/s002220050030.

[27]

C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Annals of Math., 151 (2000), 849–874. doi: 10.2307/121050.

[28]

L. X. Truong, L. T. P. Ngoc, Alain P. N. Dinh and N. T. Long, Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type, Nonlinear Anal. TMA., 74 (2011), 6933–6949. doi: 10.1016/j.na.2011.07.015.

[29]

Jieqiong Wu and Shengjia Li, Blow-up for coupled nonlinear wave equations with damping and source, Applied Mathematics Letters, 24 (2011), 1093-1098. doi: 10.1016/j.aml.2011.01.030.

[30]

Zai-yun Zhang and Xiu-jin Miao, Global existence and uniform decay for wave equation with dissipative term and boundary damping, Computers and Mathematics with Applications, 59 (2010), 1003-1018. doi: 10.1016/j.camwa.2009.09.008.

show all references

References:
[1]

M. Bergounioux, N. T. Long and Alain P. N. Dinh, Mathematical model for a shock problem involving a linear viscoelastic bar, Nonlinear Anal., 43 (2001), 547–561. doi: 10.1016/S0362-546X(99)00218-7.

[2]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho and J. A. Soriano, Existence and uniform decay of solutions of a degenerate equation with nonlinear boundary damping and boundary memory source term, Nonlinear Anal., 38 (1999), 281–294. doi: 10.1016/S0362-546X(98)00195-3.

[3]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, On the existence and the uniform decay of a hyperbolic, Southeast Asian Bulletin of Mathematics, 24 (2000), 183–199. doi: 10.1007/s100120070002.

[4]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and M. L. Santos, Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary, Applied Mathematics and Computation, 150 (2004), 439–465. doi: 10.1016/S0096-3003(03)00284-4.

[5]

Fei Liang and Hongjun Gao, Global nonexistence of positive initial-energy solutions for coupled nonlinear wave equations with damping and source terms, Abstract and Applied Analysis, Vol. 2011, Art. ID 760209, 14 pages. doi: 10.1155/2011/760209.

[6]

V. A. Khoa, L. T. P. Ngoc and N. T. Long, Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms, Evolution Equations & Control Theory, 8 (2019), 359–395.

[7]

V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Vol. 1. Academic Press, NewYork, 1969.

[8]

C. Sideris and Yi Zhou, Almost global existence for 2-D incompressible isotropic elastodynamics, Trans. Amer. Math. Soc., 367 (2015), 8175–8197. doi: 10.1090/tran/6294.

[9]

Zhen Lei, On 2D Viscoelasticity with Small Strain, Arch. Rational Mech. Anal., 198 (2010), 13-37. doi: 10.1007/s00205-010-0346-2.

[10]

J. L. Lions, Quelques méthodes de ré solution des problèmes aux limites nonlinéaires, Dunod; Gauthier-Villars, Paris, 1969.

[11]

W. Liu, G. Li and L. Hong, General decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping, Journal of Function Spaces, Vol. 2014, Art. ID 284809, 21 pages. doi: 10.1155/2014/284809.

[12]

N. T. Long and Alain P. N. Dinh, On the quasilinear wave equation: $u_tt-\Delta u+f(u, $ $u_{t}) = 0$ associated with a mixed nonhomogeneous condition, Nonlinear Anal., 19 (1992). doi: 10.1016/0362-546X(92)90097-X.

[13]

N. T. Long and Alain P. N. Dinh, A semilinear wave equation associated with a linear differential equation with Cauchy data, Nonlinear Anal. TMA., 24 (1995), 1261–1279. doi: 10.1016/0362-546X(94)00196-O.

[14]

N. T. Long and T. N. Diem, On the nonlinear wave equation $u_tt-u_xx = f(x, $ $t, \, u, $ $u_{x}, $ $u_{t})$ associated with the mixed homogeneous conditions, Nonlinear Anal. TMA., 29 (1997), 1217–1230. doi: 10.1016/S0362-546X(97)87360-9.

[15]

N. T. Long, Alain P. N. Dinh and T. N. Diem, On a shock problem involving a nonlinear viscoelastic bar, J. Boundary Value Problems, 2005 (2005), 337–358. doi: 10.1155/bvp.2005.337.

[16]

N. T. Long and L. T. P. Ngoc, On a nonlinear wave equation with boundary conditions of two-point type, J. Math. Anal. Appl., 385 (2012), 1070–1093. doi: 10.1016/j.jmaa.2011.07.034.

[17]

S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260 (2003), 58–66. doi: 10.1002/mana.200310104.

[18]

Changxing Miao and Youbin Zhu, Global smooth solutions for a nonlinear system of wave equations, Nonlinear Anal., 67 (2007), 3136-3151. doi: 10.1016/j.na.2006.10.006.

[19]

L. T. P. Ngoc and N. T. Long, Existence, blow-up and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions, Mathematical Methods in the Applied Sciences, 37 (2014), 464–487. doi: 10.1002/mma.2803.

[20]

L. T. P. Ngoc and N. T. Long, Existence and exponential decay for a nonlinear wave equation with a nonlocal boundary condition, Communications on Pure and Applied Analysis, 12 (2013), 2001–2029. doi: 10.3934/cpaa.2013.12.2001.

[21]

L. T. P. Ngoc, C. H. Hoa and N. T. Long, Existence, blow-up, and exponential decay estimates for a system of semilinear wave equations associated with the helical flows of Maxwell fluid, Mathematical Methods in the Applied Sciences, 39 (2016), 2334–2357. doi: 10.1002/mma.3643.

[22]

L. T. P. Ngoc, N. A. Triet, Alain P. N. Dinh and N. T. Long, Existence and exponential decay of solutions for a wave equation with integral nonlocal boundary conditions of memory type, Numerical Functional Analysis and Optimization, 38 (2017), 1173–1207. doi: 10.1080/01630563.2017.1320672.

[23]

L. T. P. Ngoc, L. N. K. Hang and N. T. Long, On a nonlinear wave equation associated with the boundary conditions involving convolution, Nonlinear Anal. TMA., 70 (2009), 3943–3965. doi: 10.1016/j.na.2008.08.004.

[24]

M. L. Santos, Decay rates for solutions of a system of wave equations with memory, Electronic J. Differential Equations, 38 (2002), 1–17.

[25]

C. Sideris, Global behavior of solutions to nonlinear wave equations in three dimensions, Comm. P.D.E., 8 (1983), 1291–1323. doi: 10.1080/03605308308820304.

[26]

C. Sideris, The null condition and global existence of nonlinear elastic waves, Invent. Math., 123 (1996), 323–342. doi: 10.1007/s002220050030.

[27]

C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Annals of Math., 151 (2000), 849–874. doi: 10.2307/121050.

[28]

L. X. Truong, L. T. P. Ngoc, Alain P. N. Dinh and N. T. Long, Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type, Nonlinear Anal. TMA., 74 (2011), 6933–6949. doi: 10.1016/j.na.2011.07.015.

[29]

Jieqiong Wu and Shengjia Li, Blow-up for coupled nonlinear wave equations with damping and source, Applied Mathematics Letters, 24 (2011), 1093-1098. doi: 10.1016/j.aml.2011.01.030.

[30]

Zai-yun Zhang and Xiu-jin Miao, Global existence and uniform decay for wave equation with dissipative term and boundary damping, Computers and Mathematics with Applications, 59 (2010), 1003-1018. doi: 10.1016/j.camwa.2009.09.008.

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