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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

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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

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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

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

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

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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

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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

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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

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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

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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

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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

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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

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