# American Institute of Mathematical Sciences

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

 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.

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.

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##### 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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