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Global conservative solutions for a modified periodic coupled Camassa-Holm system
On a final value problem for a nonlinear fractional pseudo-parabolic equation
1. | Division of Applied Mathematics, Thu Dau Mot University, Thu Dau Mot City, Vietnam |
2. | Department of Mathematics, University of Mazandaran, Babolsar, Iran, Department of Mathematical Sciences, University of South Africa, UNISA0003, South Africa |
3. | Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 110122, Taiwan |
4. | Department of Mathematics, FSTE Moulay Ismail University of Meknes, BP 509 Boutalamine, Errachidia 52000, Morocco |
5. | Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam |
6. | Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam, Vietnam National University, Ho Chi Minh City, Vietnam |
In this paper, we investigate a final boundary value problem for a class of fractional with parameter $ \beta $ pseudo-parabolic partial differential equations with nonlinear reaction term. For $ 0<\beta < 1, $ the solution is regularity-loss, we establish the well-posedness of solutions. In the case that $ \beta >1 $, it has a feature of regularity-gain. Then, the instability of a mild solution is proved. We introduce two methods to regularize the problem. With the help of the modified Lavrentiev regularization method and Fourier truncated regularization method, we propose the regularized solutions in the cases of globally or locally Lipschitzian source term. Moreover, the error estimates is established.
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The well-posedness and regularity of a rotating blades equation, Electron. Res. Arch., 28 (2020), 691-719.
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Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.
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Global existence and finite time blow-up of solutions for the semilinear pseudo-parabolic equation with a memory term, Appl. Anal., 98 (2019), 735-755.
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N. H. Tuan, V. V. Au, V. A. Khoa and D. Lesnic, Identification of the population density of a species model with nonlocal diffusion and nonlinear reaction, Inverse Problems, 33 (2017), 40pp.
doi: 10.1088/1361-6420/aa635f. |
[26] |
N. H. Tuan, M. Kirane, B. Samet and V. V. Au,
A new fourier truncated regularization method for semilinear backward parabolic problems, Acta Appl. Math., 148 (2017), 143-155.
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[27] |
N. H. Tuan and D. D. Trong,
A nonlinear parabolic equation backward in time: Regularization with new error estimates, Nonlinear Anal., 73 (2010), 1842-1852.
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[29] |
R. Wang, Y. Li and B. Wang,
Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.
doi: 10.3934/dcds.2019165. |
[30] |
R. Wang, L. Shi and B. Wang,
Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\mathbb R^N$, Nonlinearity, 32 (2019), 4524-4556.
doi: 10.1088/1361-6544/ab32d7. |
[31] |
R. Xu, W. Lian and Y. Niu,
Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.
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[32] |
R. Xu and J. Su,
Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.
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[33] |
R. Xu, X. 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. |
[34] |
R. Xu, M. Zhang, S. Chen, Y. Yang and J. Shen,
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Exponential growth of solution of a strongly nonlinear generalized Boussinesq equation, Comput. Math. Appl., 68 (2014), 1787-1793.
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show all references
References:
[1] |
S. Antontsev and S. Shmarev,
On a class of fully nonlinear parabolic equations, Adv. Nonlinear Anal., 8 (2019), 79-100.
doi: 10.1515/anona-2016-0055. |
[2] |
V. V. Au, M. Kirane and N. H. Tuan,
Determination of initial data for a reaction-diffusion system with variable coefficients, Discrete Contin. Dyn. Syst., 39 (2019), 771-801.
doi: 10.3934/dcds.2019032. |
[3] |
V. V. Au and N. H. Tuan,
Identification of the initial condition in backward problem with nonlinear diffusion and reaction, J. Math. Anal. Appl., 452 (2017), 176-187.
doi: 10.1016/j.jmaa.2017.02.055. |
[4] |
Y. Cao and C. Liu,
Initial boundary value problem for a mixed pseudo- parabolic $p$-Laplacian type equation with logarithmic nonlinearity, Electronic J. Differential Equations, 2018 (2018), 1-19.
|
[5] |
Y. Cao, J. Yin and C. Wang,
Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations, 246 (2009), 4568-4590.
doi: 10.1016/j.jde.2009.03.021. |
[6] |
H. Chen and S. Tian,
Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.
doi: 10.1016/j.jde.2015.01.038. |
[7] |
H. Chen and H. Xu,
Global existence and blow-up in finite time for a class of finitely degenerate semilinear pseudo-parabolic equations, Acta Math. Sin. (Engl. Ser.), 35 (2019), 1143-1162.
doi: 10.1007/s10114-019-8037-x. |
[8] |
H. Chen and H. Xu,
Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.
doi: 10.3934/dcds.2019051. |
[9] |
H. Di, Y. Shang and X. Zhang,
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. |
[10] |
H. Ding and J. Zhou,
Global existence and blow-up for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity, J. Math. Anal. Appl., 478 (2019), 393-420.
doi: 10.1016/j.jmaa.2019.05.018. |
[11] |
V. R. Gopala Rao and T. W. Ting,
Solutions of pseudo-heat equations in the whole space, Arch. Ration. Mech. Anal., 49 (1972/73), 57-78.
doi: 10.1007/BF00281474. |
[12] |
Y. He, H. Gao and H. Wang,
Blow-up and decay for a class of pseudo-parabolic $p$-Laplacian equation with logarithmic nonlinearity, Comput. Math. Appl., 75 (2018), 459-469.
doi: 10.1016/j.camwa.2017.09.027. |
[13] |
F. A. Høeg and P. Lindqvist,
Regularity of solutions of the parabolic normalized $p$-Laplace equation, Adv. Nonlinear Anal., 9 (2020), 7-15.
doi: 10.1515/anona-2018-0091. |
[14] |
L. Jin, L. Li and S. Fang,
The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation, Comput. Math. Appl., 73 (2017), 2221-2232.
doi: 10.1016/j.camwa.2017.03.005. |
[15] |
J. Johnsen,
Well-posed final value problems and Duhamel's formula for coercive Lax-Milgram operators, Electronic Res. Arch., 27 (2019), 20-36.
doi: 10.3934/era.2019008. |
[16] |
M. V. Klibanov, Carleman weight functions for solving ill-posed Cauchy problems for quasilinear PDEs, Inverse Problems, 31 (2015), 20pp.
doi: 10.1088/0266-5611/31/12/125007. |
[17] |
W. Lian, J. Wang and R. Xu,
Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.
doi: 10.1016/j.jde.2020.03.047. |
[18] |
W. Lian 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. |
[19] |
Y. Lu and L. Fei, Bounds for blow-up time in a semilinear pseudo-parabolic equation with nonlocal source, J. Inequal. Appl., 2016 (2016), 11pp.
doi: 10.1186/s13660-016-1171-4. |
[20] |
H. T. Nguyen, V. A. Khoa and V. V. Au,
Analysis of a quasi-reversibility method for a terminal value quasi-linear parabolic problem with measurements, SIAM J. Math. Anal., 51 (2019), 60-85.
doi: 10.1137/18M1174064. |
[21] |
L. Shen, S. Wang and Y. Wang,
The well-posedness and regularity of a rotating blades equation, Electron. Res. Arch., 28 (2020), 691-719.
doi: 10.3934/era.2020036. |
[22] |
R. E. Showalter and T. W. Ting,
Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.
doi: 10.1137/0501001. |
[23] |
F. Sun, L. Liu and Y. Wu,
Global existence and finite time blow-up of solutions for the semilinear pseudo-parabolic equation with a memory term, Appl. Anal., 98 (2019), 735-755.
doi: 10.1080/00036811.2017.1400536. |
[24] |
T. W. Ting,
Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan, 21 (1969), 440-453.
doi: 10.2969/jmsj/02130440. |
[25] |
N. H. Tuan, V. V. Au, V. A. Khoa and D. Lesnic, Identification of the population density of a species model with nonlocal diffusion and nonlinear reaction, Inverse Problems, 33 (2017), 40pp.
doi: 10.1088/1361-6420/aa635f. |
[26] |
N. H. Tuan, M. Kirane, B. Samet and V. V. Au,
A new fourier truncated regularization method for semilinear backward parabolic problems, Acta Appl. Math., 148 (2017), 143-155.
doi: 10.1007/s10440-016-0082-1. |
[27] |
N. H. Tuan and D. D. Trong,
A nonlinear parabolic equation backward in time: Regularization with new error estimates, Nonlinear Anal., 73 (2010), 1842-1852.
doi: 10.1016/j.na.2010.05.019. |
[28] |
R. Wang, Y. Li and B. Wang, Bi-spatial pullback attractors of fractional nonclassical diffusion equations on unbounded domains with $(p, q)$-growth nonlinearities, Appl. Math. Optim., (2020).
doi: 10.1007/s00245-019-09650-6. |
[29] |
R. Wang, Y. Li and B. Wang,
Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.
doi: 10.3934/dcds.2019165. |
[30] |
R. Wang, L. Shi and B. Wang,
Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\mathbb R^N$, Nonlinearity, 32 (2019), 4524-4556.
doi: 10.1088/1361-6544/ab32d7. |
[31] |
R. Xu, W. Lian and Y. Niu,
Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.
doi: 10.1007/s11425-017-9280-x. |
[32] |
R. Xu and J. Su,
Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.
doi: 10.1016/j.jfa.2013.03.010. |
[33] |
R. Xu, X. 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. |
[34] |
R. Xu, M. Zhang, S. Chen, Y. 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. |
[35] |
H. Zhang, J. Lu and Q. Hu,
Exponential growth of solution of a strongly nonlinear generalized Boussinesq equation, Comput. Math. Appl., 68 (2014), 1787-1793.
doi: 10.1016/j.camwa.2014.10.012. |
[36] |
X. Zhu, F. Li and Y. Li,
Global solutions and blow up solutions to a class of pseudo-parabolic equations with nonlocal term, Appl. Math. Comput., 329 (2018), 38-51.
doi: 10.1016/j.amc.2018.02.003. |
[37] |
X. Zhu, F. Li, Z. Liang and T. Rong,
A sufficient condition for blowup of solutions to a class of pseudo-parabolic equations with a nonlocal term, Math. Methods Appl. Sci., 39 (2016), 3591-3606.
doi: 10.1002/mma.3803. |
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