American Institute of Mathematical Sciences

Advanced
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, 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]

Pao-Liu Chow. Asymptotic solutions of a nonlinear stochastic beam equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 735-749. doi: 10.3934/dcdsb.2006.6.735
[2]

Abhishake Rastogi. Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4111-4126. doi: 10.3934/cpaa.2020183
[3]

Tarek Saanouni. Remarks on the damped nonlinear Schrödinger equation. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020030
[4]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[5]

S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593
[6]

Thorsten Hohage, Mihaela Pricop. Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Problems & Imaging, 2008, 2 (2) : 271-290. doi: 10.3934/ipi.2008.2.271
[7]

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
[8]

Davit Martirosyan. Exponential mixing for the white-forced damped nonlinear wave equation. Evolution Equations & Control Theory, 2014, 3 (4) : 645-670. doi: 10.3934/eect.2014.3.645
[9]

Hiroshi Takeda. Large time behavior of solutions for a nonlinear damped wave equation. Communications on Pure & Applied Analysis, 2016, 15 (1) : 41-55. doi: 10.3934/cpaa.2016.15.41
[10]

Olivier Goubet, Ezzeddine Zahrouni. On a time discretization of a weakly damped forced nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1429-1442. doi: 10.3934/cpaa.2008.7.1429
[11]

Dongsheng Yin, Min Tang, Shi Jin. The Gaussian beam method for the wigner equation with discontinuous potentials. Inverse Problems & Imaging, 2013, 7 (3) : 1051-1074. doi: 10.3934/ipi.2013.7.1051
[12]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387
[13]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367
[14]

Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307
[15]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064
[16]

Yanan Li, Zhijian Yang, Fang Da. Robust attractors for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5975-6000. doi: 10.3934/dcds.2019261
[17]

Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002
[18]

Fuqin Sun, Mingxin Wang. Non-existence of global solutions for nonlinear strongly damped hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 949-958. doi: 10.3934/dcds.2005.12.949
[19]

Jiao Chen, Weike Wang. The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 307-330. doi: 10.3934/cpaa.2014.13.307
[20]

Olivier Goubet. Approximate inertial manifolds for a weakly damped nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 503-530. doi: 10.3934/dcds.1997.3.503

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (30)
  • HTML views (22)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences