# American Institute of Mathematical Sciences

March  2021, 29(1): 1709-1734. doi: 10.3934/era.2020088

## 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

* Corresponding author: Nguyen Huy Tuan

Received  May 2020 Revised  July 2020 Published  March 2021 Early access  August 2020

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.

Citation: Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088
##### References:
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Differential Equations, 246 (2009), 4568-4590.  doi: 10.1016/j.jde.2009.03.021.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] L. Jin, L. Li and S. 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Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.  doi: 10.1137/0501001.  Google Scholar [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.  Google Scholar [24] T. W. Ting, Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan, 21 (1969), 440-453.  doi: 10.2969/jmsj/02130440.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.  doi: 10.1137/0501001.  Google Scholar [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.  Google Scholar [24] T. W. Ting, Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan, 21 (1969), 440-453.  doi: 10.2969/jmsj/02130440.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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