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Boundary null-controllability of coupled parabolic systems with Robin conditions
On a final value problem for a class of nonlinear hyperbolic equations with damping term
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 |
$ u(x,t) $ |
$ (x,t)\in \Omega \times [0,T] $ |
$ u(x,T) = g(x) $ |
$ u_t(x,T) = {h(x)} $ |
$ u_{tt} + a \Delta^2 u_t + b \Delta^2 u = \mathcal R(u). $ |
$ \mathcal R $ |
$ \mathcal R(u) = u|u|^{p-1}, p>1 $ |
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. |
[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. |
[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. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[10] |
G. Chen,
Initial boundary value problem for a damped nonlinear hyperbolic equation, J. Partial Differential Equations, 16 (2003), 49-61.
|
[11] |
D. Henry, Geometric theory of semilinear parabolic equations, in Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[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. |
[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. |
[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. |
[15] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
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. |
[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. |
[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. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[10] |
G. Chen,
Initial boundary value problem for a damped nonlinear hyperbolic equation, J. Partial Differential Equations, 16 (2003), 49-61.
|
[11] |
D. Henry, Geometric theory of semilinear parabolic equations, in Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[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. |
[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. |
[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. |
[15] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
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