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March  2020, 28(1): 221-261. doi: 10.3934/era.2020015

Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source

1. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

2. 

School of Science, Dalian Jiaotong University, Dalian 116028, China

* Corresponding author: Huafei Di, dihuafei@gzhu.edu.cn

Received  October 2019 Revised  January 2020

Fund Project: Huafei Di is supported by the NSF of China (11801108, 11701116), the Scientific Program (2016A030310262) of Guangdong Province, and the College Scientific Research Project (1201630180) of Guangzhou City

In this paper, we consider the initial boundary value problem for the fourth order wave equation with nonlinear boundary velocity feedbacks $ f_{1}(u_{\nu{t}}) $, $ f_{2}(u_{t}) $ and internal source $ |u|^{\rho}u $. Under some geometrical conditions, the existence and uniform decay rates of the solutions are proved even if the nonlinear boundary velocity feedbacks $ f_{1}(u_{\nu{t}}) $, $ f_{2}(u_{t}) $ have not polynomial growth near the origin respectively. By the combination of the Galerkin approximation, potential well method and a special basis constructed, we first obtain the global existence and uniqueness of regular solutions and weak solutions. In addition, we also investigate the explicit decay rate estimates of the energy, the ideas of which are based on the construction of a special weight function $ \phi(t) $ (that depends on the behaviors of the functions $ f_{1}(u_{\nu{t}}) $, $ f_{2}(u_{t}) $ near the origin), nonlinear integral inequality and the Multiplier method.

Citation: Huafei Di, Yadong Shang, Jiali Yu. Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source. Electronic Research Archive, 2020, 28 (1) : 221-261. doi: 10.3934/era.2020015
References:
[1]

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H. DiY. Shang and X. Zheng, Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 781-801.  doi: 10.3934/dcdsb.2016.21.781.  Google Scholar

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H. Di and Y. Shang, Global existence and nonexistence of solutions for the nonlinear pseudo-parabolic equation with a memory term, Math. Methods Appl. Sci., 38 (2015), 3923-3936.  doi: 10.1002/mma.3327.  Google Scholar

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[14]

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[19]

I. Lasiecka, Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. Differential Equations, 79 (1989), 340-381.  doi: 10.1016/0022-0396(89)90107-1.  Google Scholar

[20]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533.   Google Scholar

[21]

P. D. LaxC. S. Morawetz and R. S. Phllips, The exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle, Bull. Amer. Math. Soc., 68 (1962), 593-595.  doi: 10.1090/S0002-9904-1962-10865-9.  Google Scholar

[22]

H. A. Levine and R. A. Smith, A potential well theory for the wave equation with a nonlinear boundary condition, J. Reine Angew. Math., 374 (1987), 1-23.  doi: 10.1515/crll.1987.374.1.  Google Scholar

[23]

J.-L. Lions, Quelques Méthodes de Résolution des Problèms aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[24]

W. Liu, Y. Sun and G. Li, Blow-up solutions for a nonlinear wave equation with nonnegative initial energy, Electron. J. Differential Equations, 115 (2013), 8 pp.  Google Scholar

[25]

W. Liu and J. Yu, On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms, Nonlinear Anal., 74 (2011), 2175-2190.  doi: 10.1016/j.na.2010.11.022.  Google Scholar

[26]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var., 4 (1999), 419-444.  doi: 10.1051/cocv:1999116.  Google Scholar

[27]

S. A. Messaoudi, Global nonexistence in a nonlinearly damped wave equation, Appl. Anal., 80 (2001), 269-277.  doi: 10.1080/00036810108840993.  Google Scholar

[28]

J. P. Puel and M. Tucsnak, Stabilisation frontière pour les équations de von Kármán, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 609-612.   Google Scholar

[29]

A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl., 71 (1992), 455-467.   Google Scholar

[30]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942.  Google Scholar

[31]

E. Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasg. Math. J., 44 (2002), 375-395.  doi: 10.1017/S0017089502030045.  Google Scholar

[32]

E. Vitillaro, Some new results on global nonexistence and blow-up for evolution problems with positive initial energy, Rend. Istit. Mat. Univ. Trieste, 31 (2000), 245-275.   Google Scholar

[33]

E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298.  doi: 10.1016/S0022-0396(02)00023-2.  Google Scholar

[34]

L. Wei and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

[35]

R. XuW. LianX. Kong and Y. Yang, Fourth order wave equation with nonlinear strain and logarithmic nonlinearity, Appl. Numer. Math., 141 (2019), 185-205.  doi: 10.1016/j.apnum.2018.06.004.  Google Scholar

[36]

R. XuX. Wang and Y. Yang, Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Appl. Math. Lett., 83 (2018), 176-181.  doi: 10.1016/j.aml.2018.03.033.  Google Scholar

[37]

R. Xu and Y. Yang, Finite time blow-up for the nonlinear fourth-order dispersive-dissipative wave equation at high energy level, Internat. J. Math., 23 (2012), 1250060, 10 pp. doi: 10.1142/S0129167X12500607.  Google Scholar

[38]

R. XuY. Yang and Y. Liu, Global well-posedness for strongly damped viscoelastic wave equation, Appl. Anal., 92 (2013), 138-157.  doi: 10.1080/00036811.2011.601456.  Google Scholar

[39]

R. Xu, X. Wang, Y. Yang and S. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503, 27 pp. doi: 10.1063/1.5006728.  Google Scholar

[40]

R. XuM. ZhangS. ChenY. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

[41]

Y. Yang and R. Xu, Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up, Commun. Pure Appl. Anal., 18 (2019), 1351-1358.  doi: 10.3934/cpaa.2019065.  Google Scholar

[42]

H. Zhang and Q. Hu, Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition, Commun. Pure Appl. Anal., 4 (2005), 861-869.  doi: 10.3934/cpaa.2005.4.861.  Google Scholar

[43]

E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim., 28 (1990), 466-478.  doi: 10.1137/0328025.  Google Scholar

show all references

References:
[1]

A. Benaissa and S. A. Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 391-399.  doi: 10.1007/s00030-005-0008-5.  Google Scholar

[2]

L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860.  doi: 10.3934/dcds.2008.22.835.  Google Scholar

[3]

M. M. CavalcantiV. N. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differential Equations, 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004.  Google Scholar

[4]

M. M. CavalcantiV. N. D. Cavalcanti and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203 (2004), 119-158.  doi: 10.1016/j.jde.2004.04.011.  Google Scholar

[5]

G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl., 58 (1979), 249-273.   Google Scholar

[6]

H. DiY. Shang and X. Zheng, Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 781-801.  doi: 10.3934/dcdsb.2016.21.781.  Google Scholar

[7]

H. Di and Y. Shang, Global existence and nonexistence of solutions for the nonlinear pseudo-parabolic equation with a memory term, Math. Methods Appl. Sci., 38 (2015), 3923-3936.  doi: 10.1002/mma.3327.  Google Scholar

[8]

H. Di and Y. Shang, Global existence and asymptotic behavior of solutions for the double dispersive-dissipative wave equation with nonlinear damping and source terms, Bound. Value Probl., 2015 (2015), 15 pp. doi: 10.1186/s13661-015-0288-6.  Google Scholar

[9]

H. Di and Y. Shang, Existence, nonexistence and decay estimate of global solutions for a viscoelastic wave equation with nonlinear boundary damping and internal source terms, Eur. J. Pure Appl. Math., 10 (2017), 668-701.   Google Scholar

[10]

L. C. Evance, Partial Differential Equations, Second edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[11]

H. FengS. Li and X. Zhi, Blow-up solutions for a nonlinear wave equation with boundary damping and interior source, Nonlinear Anal., 75 (2012), 2273-2280.  doi: 10.1016/j.na.2011.10.027.  Google Scholar

[12]

H. Feng and S. Li, Global nonexistence for a semilinear wave equation with nonlinear boundary dissipation, J. Math. Anal. Appl., 391 (2012), 255-264.  doi: 10.1016/j.jmaa.2012.02.013.  Google Scholar

[13]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051.  Google Scholar

[14]

A. Guesmia, Existence globale et stabilisation interne non linéaire d'un système de Petrovsky, Bull. Belg. Math. Soc. Simon Stevin, 5 (1998), 583-594.  doi: 10.36045/bbms/1103309996.  Google Scholar

[15]

R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal., 27 (1996), 1165-1175.  doi: 10.1016/0362-546X(95)00119-G.  Google Scholar

[16]

V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar

[17]

J. Lagnese and J.-L. Lions, Modelling Analysis and Control of Thin Plates, Research in Applied Mathematics, Masson, Paris, 1988.  Google Scholar

[18]

J. E. Lagnese, Boundary Stabilization of Thin Plates, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[19]

I. Lasiecka, Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. Differential Equations, 79 (1989), 340-381.  doi: 10.1016/0022-0396(89)90107-1.  Google Scholar

[20]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533.   Google Scholar

[21]

P. D. LaxC. S. Morawetz and R. S. Phllips, The exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle, Bull. Amer. Math. Soc., 68 (1962), 593-595.  doi: 10.1090/S0002-9904-1962-10865-9.  Google Scholar

[22]

H. A. Levine and R. A. Smith, A potential well theory for the wave equation with a nonlinear boundary condition, J. Reine Angew. Math., 374 (1987), 1-23.  doi: 10.1515/crll.1987.374.1.  Google Scholar

[23]

J.-L. Lions, Quelques Méthodes de Résolution des Problèms aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[24]

W. Liu, Y. Sun and G. Li, Blow-up solutions for a nonlinear wave equation with nonnegative initial energy, Electron. J. Differential Equations, 115 (2013), 8 pp.  Google Scholar

[25]

W. Liu and J. Yu, On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms, Nonlinear Anal., 74 (2011), 2175-2190.  doi: 10.1016/j.na.2010.11.022.  Google Scholar

[26]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var., 4 (1999), 419-444.  doi: 10.1051/cocv:1999116.  Google Scholar

[27]

S. A. Messaoudi, Global nonexistence in a nonlinearly damped wave equation, Appl. Anal., 80 (2001), 269-277.  doi: 10.1080/00036810108840993.  Google Scholar

[28]

J. P. Puel and M. Tucsnak, Stabilisation frontière pour les équations de von Kármán, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 609-612.   Google Scholar

[29]

A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl., 71 (1992), 455-467.   Google Scholar

[30]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942.  Google Scholar

[31]

E. Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasg. Math. J., 44 (2002), 375-395.  doi: 10.1017/S0017089502030045.  Google Scholar

[32]

E. Vitillaro, Some new results on global nonexistence and blow-up for evolution problems with positive initial energy, Rend. Istit. Mat. Univ. Trieste, 31 (2000), 245-275.   Google Scholar

[33]

E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298.  doi: 10.1016/S0022-0396(02)00023-2.  Google Scholar

[34]

L. Wei and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

[35]

R. XuW. LianX. Kong and Y. Yang, Fourth order wave equation with nonlinear strain and logarithmic nonlinearity, Appl. Numer. Math., 141 (2019), 185-205.  doi: 10.1016/j.apnum.2018.06.004.  Google Scholar

[36]

R. XuX. Wang and Y. Yang, Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Appl. Math. Lett., 83 (2018), 176-181.  doi: 10.1016/j.aml.2018.03.033.  Google Scholar

[37]

R. Xu and Y. Yang, Finite time blow-up for the nonlinear fourth-order dispersive-dissipative wave equation at high energy level, Internat. J. Math., 23 (2012), 1250060, 10 pp. doi: 10.1142/S0129167X12500607.  Google Scholar

[38]

R. XuY. Yang and Y. Liu, Global well-posedness for strongly damped viscoelastic wave equation, Appl. Anal., 92 (2013), 138-157.  doi: 10.1080/00036811.2011.601456.  Google Scholar

[39]

R. Xu, X. Wang, Y. Yang and S. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503, 27 pp. doi: 10.1063/1.5006728.  Google Scholar

[40]

R. XuM. ZhangS. ChenY. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

[41]

Y. Yang and R. Xu, Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up, Commun. Pure Appl. Anal., 18 (2019), 1351-1358.  doi: 10.3934/cpaa.2019065.  Google Scholar

[42]

H. Zhang and Q. Hu, Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition, Commun. Pure Appl. Anal., 4 (2005), 861-869.  doi: 10.3934/cpaa.2005.4.861.  Google Scholar

[43]

E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim., 28 (1990), 466-478.  doi: 10.1137/0328025.  Google Scholar

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