American Institute of Mathematical Sciences

Advanced
March  2021, 10(1): 103-127. doi: 10.3934/eect.2020053

On a final value problem for a class of nonlinear hyperbolic equations with damping term

Nguyen Huu Can 1, , Nguyen Huy Tuan 1, , Donal O'Regan 2, and  Vo Van Au 3,,

1. 

Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam, Vietnam National University, Ho Chi Minh City, Vietnam
2. 

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
3. 

Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh City 700000, Vietnam, Faculty of Natural Sciences, Duy Tan University, Da Nang, 550000, Vietnam

* Corresponding author: vovanau@duytan.edu.vn (Vo Van Au)

Received  December 2019 Revised  February 2020 Published  May 2020

Fund Project: The second author is supported by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number B2020-18-03

This paper deals with the problem of finding the function
$ u(x,t) $
,
$ (x,t)\in \Omega \times [0,T] $
, from the final data
$ u(x,T) = g(x) $
 and
$ u_t(x,T) = {h(x)} $
,
$ u_{tt} + a \Delta^2 u_t + b \Delta^2 u = \mathcal R(u). $
This problem is known as the inverse initial problem for the nonlinear hyperbolic equation with damping term and it is ill-posed in the sense of Hadamard. In order to stabilize the solution, we propose the filter regularization method to regularize the solution. We establish appropriate filtering functions in cases where the nonlinear source
$ \mathcal R $
 satisfies the global Lipschitz condition and the specific case
$ \mathcal R(u) = u|u|^{p-1}, p>1 $
 which satisfies the local Lipschitz condition. In addition, we show that regularized solutions converge to the sought solution under a priori assumptions in Gevrey spaces.
Keywords: Inverse problems, nonlinear damped hyperbolic equation, nonlinear beam equation, regularization method, error estimate.
Mathematics Subject Classification: 35R30, 35L35, 47J06, 47H10, 47A52.
Citation: Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053
References:
[1]

M. Aassila and A. Guesmia, Energy decay for a damped nonlinear hyperbolic equation, Appl. Math. Lett., 12 (1999), 49-52.  doi: 10.1016/S0893-9659(98)00171-2.  Google Scholar

[2]

A. S. Ackleh, H. T. Banks and G. A. Pinter, A nonlinear beam equation, Appl. Math. Lett., 15 (2002), 381-387.  doi: 10.1016/S0893-9659(01)00147-1.  Google Scholar

[3]

R. P. Agarwal, S. Hodis and D. O'Regan, 500 Examples and Problems of Applied Differential Equations, Problem Books in Mathematics, Springer, Cham, 2019. doi: 10.1007/978-3-030-26384-3.  Google Scholar

[4]

H. T. Banks, K. Ito and Y. Wang, Well posedness for damped second-order systems with unbounded input operators, Differential Integral Equations, 8 (1995), 587-606.   Google Scholar

[5]

H. T. Banks, D. S. Gilliam and V. I. Shubov, Global solvability for damped abstract nonlinear hyperbolic systems, Differential Integral Equations, 10 (1997), 309-332.   Google Scholar

[6]

C. Cao, M. A. Rammaha and E. S. Titi, The Navier-Stokes equations on the rotating $2$-D sphere: Gevrey regularity and asymptotic degrees of freedom, Z. Angew. Math. Phys., 50 (1999), 341-360.  doi: 10.1007/PL00001493.  Google Scholar

[7]

G. Chen and B. Lu, The initial-boundary value problems for a class of nonlinear wave equations with damping term, J. Math. Anal. Appl., 351 (2009), 1-15.  doi: 10.1016/j.jmaa.2008.08.027.  Google Scholar

[8]

G. Chen and F. Da, Blow-up of solution of Cauchy problem for three-dimensional damped nonlinear hyperbolic equation, Nonlinear Anal., 71 (2009), 358-372.  doi: 10.1016/j.na.2008.10.132.  Google Scholar

[9]

G. Chen, Y. Wang and Z. Zhao, Blow-up of solution of an initial boundary value problem for a damped nonlinear hyperbolic equation, Appl. Math. Lett., 17 (2004), 491-497.  doi: 10.1016/S0893-9659(04)90116-4.  Google Scholar

[10]

G. Chen, Initial boundary value problem for a damped nonlinear hyperbolic equation, J. Partial Differential Equations, 16 (2003), 49-61.   Google Scholar

[11]

D. Henry, Geometric theory of semilinear parabolic equations, in Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[12]

T. Hosonoa and T. Ogawa, Large time behavior and $L^p$-$L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differential Equations, 203 (2004), 82-118.  doi: 10.1016/j.jde.2004.03.034.  Google Scholar

[13]

B. Jin, B. Li and Z. Zhou, Numerical analysis of nonlinear subdiffusion equations, SIAM J. Numer. Anal., 56 (2018), 1-23.  doi: 10.1137/16M1089320.  Google Scholar

[14]

W. Liu and K. Chen, Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions, Math. Nachr., 289 (2016), 300-320.  doi: 10.1002/mana.201400343.  Google Scholar

[15] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar
[16]

T. Narazaki, $L^p$-$L^q$ estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585-626.  doi: 10.2969/jmsj/1191418647.  Google Scholar

[17]

H. T. Nguyen, V. N. Doan, V. A. Khoa and V. A. Vo, A note on the derivation of filter regularization operators for nonlinear evolution equations, Appl. Anal., 97 (2018), 3-12.  doi: 10.1080/00036811.2016.1276176.  Google Scholar

[18]

K. Nishihara, $L^p$-$L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z., 244 (2003), 631-649.  doi: 10.1007/s00209-003-0516-0.  Google Scholar

[19]

T. Ogawa and H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal., 70 (2009), 3696-3701.  doi: 10.1016/j.na.2008.07.025.  Google Scholar

[20]

C. Song and Z. Yang, Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation, Math. Methods Appl. Sci., 33 (2010), 563-575.  doi: 10.1002/mma.1175.  Google Scholar

[21]

H. Takeda, Global existence and nonexistence of solutions for a system of nonlinear damped wave equations, J. Math. Anal. Appl., 360 (2009), 631-650.  doi: 10.1016/j.jmaa.2009.06.072.  Google Scholar

[22]

N. H. Tuan, D. T. Dang, E. Nane and D. M. Nguyen, Continuity of solutions of a class of fractional equations, Potential Anal., 49 (2018), 423-478.  doi: 10.1007/s11118-017-9663-5.  Google Scholar

[23]

Y.-Z. Wang, Asymptotic behavior of solutions to the damped nonlinear hyperbolic equation, J. Appl. Math., 2013, Art. ID 353757, 8 pp. doi: 10.1155/2013/353757.  Google Scholar

[24]

Z. Yang, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations, 187 (2003), 520-540.   Google Scholar

[25]

Z. Yang, Global existence, asymptotic behavior and blowup of solutions to a nonlinear evolution equation, Acta Anal. Funct. Appl., 4 (2002), 350-356.   Google Scholar

[26]

J. Yu, Y. Shang and H. Di, On decay and blow-up of solutions for a nonlinear beam equation with double damping terms, Bound. Value Probl., 145 (2018), 17 pp. doi: 10.1186/s13661-018-1067-y.  Google Scholar

[27]

J. Yu, Y. Shang and H. Di, Existence and nonexistence of global solutions to the Cauchy problem of the nonlinear hyperbolic equation with damping term, AIMS Mathematics, 3 (2018), 322-342.   Google Scholar

[28]

Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris Sér I Math., 333 (2001), 109-114.  doi: 10.1016/S0764-4442(01)01999-1.  Google Scholar

show all references

References:
[1]

M. Aassila and A. Guesmia, Energy decay for a damped nonlinear hyperbolic equation, Appl. Math. Lett., 12 (1999), 49-52.  doi: 10.1016/S0893-9659(98)00171-2.  Google Scholar

[2]

A. S. Ackleh, H. T. Banks and G. A. Pinter, A nonlinear beam equation, Appl. Math. Lett., 15 (2002), 381-387.  doi: 10.1016/S0893-9659(01)00147-1.  Google Scholar

[3]

R. P. Agarwal, S. Hodis and D. O'Regan, 500 Examples and Problems of Applied Differential Equations, Problem Books in Mathematics, Springer, Cham, 2019. doi: 10.1007/978-3-030-26384-3.  Google Scholar

[4]

H. T. Banks, K. Ito and Y. Wang, Well posedness for damped second-order systems with unbounded input operators, Differential Integral Equations, 8 (1995), 587-606.   Google Scholar

[5]

H. T. Banks, D. S. Gilliam and V. I. Shubov, Global solvability for damped abstract nonlinear hyperbolic systems, Differential Integral Equations, 10 (1997), 309-332.   Google Scholar

[6]

C. Cao, M. A. Rammaha and E. S. Titi, The Navier-Stokes equations on the rotating $2$-D sphere: Gevrey regularity and asymptotic degrees of freedom, Z. Angew. Math. Phys., 50 (1999), 341-360.  doi: 10.1007/PL00001493.  Google Scholar

[7]

G. Chen and B. Lu, The initial-boundary value problems for a class of nonlinear wave equations with damping term, J. Math. Anal. Appl., 351 (2009), 1-15.  doi: 10.1016/j.jmaa.2008.08.027.  Google Scholar

[8]

G. Chen and F. Da, Blow-up of solution of Cauchy problem for three-dimensional damped nonlinear hyperbolic equation, Nonlinear Anal., 71 (2009), 358-372.  doi: 10.1016/j.na.2008.10.132.  Google Scholar

[9]

G. Chen, Y. Wang and Z. Zhao, Blow-up of solution of an initial boundary value problem for a damped nonlinear hyperbolic equation, Appl. Math. Lett., 17 (2004), 491-497.  doi: 10.1016/S0893-9659(04)90116-4.  Google Scholar

[10]

G. Chen, Initial boundary value problem for a damped nonlinear hyperbolic equation, J. Partial Differential Equations, 16 (2003), 49-61.   Google Scholar

[11]

D. Henry, Geometric theory of semilinear parabolic equations, in Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[12]

T. Hosonoa and T. Ogawa, Large time behavior and $L^p$-$L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differential Equations, 203 (2004), 82-118.  doi: 10.1016/j.jde.2004.03.034.  Google Scholar

[13]

B. Jin, B. Li and Z. Zhou, Numerical analysis of nonlinear subdiffusion equations, SIAM J. Numer. Anal., 56 (2018), 1-23.  doi: 10.1137/16M1089320.  Google Scholar

[14]

W. Liu and K. Chen, Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions, Math. Nachr., 289 (2016), 300-320.  doi: 10.1002/mana.201400343.  Google Scholar

[15] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar
[16]

T. Narazaki, $L^p$-$L^q$ estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585-626.  doi: 10.2969/jmsj/1191418647.  Google Scholar

[17]

H. T. Nguyen, V. N. Doan, V. A. Khoa and V. A. Vo, A note on the derivation of filter regularization operators for nonlinear evolution equations, Appl. Anal., 97 (2018), 3-12.  doi: 10.1080/00036811.2016.1276176.  Google Scholar

[18]

K. Nishihara, $L^p$-$L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z., 244 (2003), 631-649.  doi: 10.1007/s00209-003-0516-0.  Google Scholar

[19]

T. Ogawa and H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal., 70 (2009), 3696-3701.  doi: 10.1016/j.na.2008.07.025.  Google Scholar

[20]

C. Song and Z. Yang, Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation, Math. Methods Appl. Sci., 33 (2010), 563-575.  doi: 10.1002/mma.1175.  Google Scholar

[21]

H. Takeda, Global existence and nonexistence of solutions for a system of nonlinear damped wave equations, J. Math. Anal. Appl., 360 (2009), 631-650.  doi: 10.1016/j.jmaa.2009.06.072.  Google Scholar

[22]

N. H. Tuan, D. T. Dang, E. Nane and D. M. Nguyen, Continuity of solutions of a class of fractional equations, Potential Anal., 49 (2018), 423-478.  doi: 10.1007/s11118-017-9663-5.  Google Scholar

[23]

Y.-Z. Wang, Asymptotic behavior of solutions to the damped nonlinear hyperbolic equation, J. Appl. Math., 2013, Art. ID 353757, 8 pp. doi: 10.1155/2013/353757.  Google Scholar

[24]

Z. Yang, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations, 187 (2003), 520-540.   Google Scholar

[25]

Z. Yang, Global existence, asymptotic behavior and blowup of solutions to a nonlinear evolution equation, Acta Anal. Funct. Appl., 4 (2002), 350-356.   Google Scholar

[26]

J. Yu, Y. Shang and H. Di, On decay and blow-up of solutions for a nonlinear beam equation with double damping terms, Bound. Value Probl., 145 (2018), 17 pp. doi: 10.1186/s13661-018-1067-y.  Google Scholar

[27]

J. Yu, Y. Shang and H. Di, Existence and nonexistence of global solutions to the Cauchy problem of the nonlinear hyperbolic equation with damping term, AIMS Mathematics, 3 (2018), 322-342.   Google Scholar

[28]

Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris Sér I Math., 333 (2001), 109-114.  doi: 10.1016/S0764-4442(01)01999-1.  Google Scholar

[1]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
[2]

Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012
[3]

Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019
[4]

Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021030
[5]

Jie-Wen He, Chi-Chon Lei, Chen-Yang Shi, Seak-Weng Vong. An inexact alternating direction method of multipliers for a kind of nonlinear complementarity problems. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 353-362. doi: 10.3934/naco.2020030
[6]

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, 2021, 15 (3) : 415-443. doi: 10.3934/ipi.2020074
[7]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448
[8]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021
[9]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028
[10]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031
[11]

Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021043
[12]

Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021055
[13]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241
[14]

Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021057
[15]

Caichun Chai, Tiaojun Xiao, Zhangwei Feng. Evolution of revenue preference for competing firms with nonlinear inverse demand. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021071
[16]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002
[17]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2601-2617. doi: 10.3934/dcds.2020376
[18]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021032
[19]

Seddigheh Banihashemi, Hossein Jafaria, Afshin Babaei. A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021025
[20]

Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019

2019 Impact Factor: 0.953

Tools

Metrics

  • PDF downloads (303)
  • HTML views (311)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences