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February  2021, 20(2): 583-621. doi: 10.3934/cpaa.2020282

## Semilinear Caputo time-fractional pseudo-parabolic equations

 1 Department of Mathematics and Computer Science, University of Science Ho Chi Minh City, Vietnam 2 Vietnam National University, Ho Chi Minh City, Vietnam 3 Division of Applied Mathematics, Thu Dau Mot University Binh Duong Province, Vietnam 4 Institute of Fundamental and Applied Sciences, Duy Tan University Ho Chi Minh City, 700000, Vietnam 5 Faculty of Natural Sciences, Duy Tan University, Da Nang, 550000, Vietnam 6 College of Mathematical Sciences, Harbin Engineering University, 150001, China

*Corresponding author

Received  July 2020 Revised  September 2020 Published  December 2020

Fund Project: The first and the second author were supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2019.09. The third author was supported by National Natural Science Foundation of China (11871017)

This paper considers two problems: the initial boundary value problem of nonlinear Caputo time-fractional pseudo-parabolic equations with fractional Laplacian, and the Cauchy problem (initial value problem) of Caputo time-fractional pseudo-parabolic equations. For the first problem with the source term satisfying the globally Lipschitz condition, we establish the local well-posedness theory including existence, uniqueness and regularity of the local solution, and the further local existence theory related to the finite time blow-up are also obtained for the problem with logarithmic nonlinearity. For the second problem with the source term satisfying the globally Lipschitz condition, we prove the global existence theorem.

Citation: Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 583-621. doi: 10.3934/cpaa.2020282
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##### References:
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Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.  Google Scholar [35] S. Ji, J. Yin and Y. Cao, Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equ., 261 (2016), 5446-5464.  doi: 10.1016/j.jde.2016.08.017.  Google Scholar [36] W. Lian, J. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differ. Equ., 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.  Google Scholar [37] A. Magana, A. Miranville and R. Quintanilla, On the time decay in phase-lag thermoelasticity with two temperatures, Electron. Res. Arch., 27 (2019), 7-19.  doi: 10.3934/era.2019007.  Google Scholar [38] B. B. Mandelbrot and J. W. V. Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093.  Google Scholar [39] E. Di Nezza, G. Palatucci and E. 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