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

Nguyen Thanh Long; Hoang Hai Ha; Le Thi Phuong Ngoc; Nguyen Anh Triet

doi:10.3934/cpaa.2020023

Previous Article
The weak maximum principle for second-order elliptic and parabolic conormal derivative problems
CPAA Home
This Issue
Next Article
On the Schrödinger-Debye system in compact Riemannian manifolds

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 & 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

[1]	Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671
[2]	Mengxian Lv, Jianghao Hao. General decay and blow-up for coupled Kirchhoff wave equations with dynamic boundary conditions. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021058
[3]	Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020
[4]	Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[5]	Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure & Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521
[6]	Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54
[7]	Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[8]	Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[9]	Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683
[10]	Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370
[11]	Vo Anh Khoa, Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms. Evolution Equations & Control Theory, 2019, 8 (2) : 359-395. doi: 10.3934/eect.2019019
[12]	Mohammad Kafini. On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021093
[13]	Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations & Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669
[14]	Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771
[15]	Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677
[16]	Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027
[17]	Van Duong Dinh. Blow-up criteria for linearly damped nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2021, 10 (3) : 599-617. doi: 10.3934/eect.2020082
[18]	Wenjun Liu, Jiangyong Yu, Gang Li. Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4337-4366. doi: 10.3934/dcdss.2021121
[19]	Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001
[20]	Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28 (1) : 263-289. doi: 10.3934/era.2020016

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors