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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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