# American Institute of Mathematical Sciences

doi: 10.3934/eect.2020109
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## On time fractional pseudo-parabolic equations with nonlocal integral conditions

 1 Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam 2 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland 3 Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey, Institute of Space Sciences, P.O.Box, MG-23, R 76900, Magurele-Bucharest, Romania

* Corresponding author: Nguyen H. Tuan

Received  July 2020 Revised  October 2020 Early access December 2020

In this paper, we study the nonlocal problem for pseudo-parabolic equation with time and space fractional derivatives. The time derivative is of Caputo type and of order $\sigma,\; \; 0<\sigma<1$ and the space fractional derivative is of order $\alpha,\beta >0$. In the first part, we obtain some results of the existence and uniqueness of our problem with suitably chosen $\alpha, \beta$. The technique uses a Sobolev embedding and is based on constructing a Mittag-Leffler operator. In the second part, we give the ill-posedness of our problem and give a regularized solution. An error estimate in $L^p$ between the regularized solution and the sought solution is obtained.

Citation: Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, doi: 10.3934/eect.2020109
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