# American Institute of Mathematical Sciences

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, doi: 10.3934/cpaa.2020282
##### References:
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##### References:
 [1] J. M. Arrieta and A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, T. Am. Math. Soc., 352 (1999), 285-310.  doi: 10.1090/S0002-9947-99-02528-3.  Google Scholar [2] R. P. Agarwal, M. Meehan and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511543005.  Google Scholar [3] G. Akagi, Fractional flows driven by subdifferentials in Hilbert spaces, Israel J. Math., 234 (2019), 809-862.  doi: 10.1007/s11856-019-1936-9.  Google Scholar [4] B. de Andrade and A. Viana, Abstract Volterra integro-differential equations with applications to parabolic models with memory, Math. Ann., 369 (2017), 1131-1175.  doi: 10.1007/s00208-016-1469-z.  Google Scholar [5] B. Andrade and A. Viana, On a fractional reaction-diffusion equation, Z. Angew. Math. Phys., 68 (2017), 11 pp. doi: 10.1007/s00033-017-0801-0.  Google Scholar [6] S. Antontsev and S. 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