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On the well-posedness of a nonlinear pseudo-parabolic equation

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Abstract

In this paper we consider the Cauchy problem for the pseudo-parabolic equation:

$$\begin{aligned} \dfrac{\partial }{\partial t} \left( u + \mu (-\Delta )^{s_1} u\right) + (-\Delta )^{s_2} u = f(u),\quad x \in \Omega ,~ t>0. \end{aligned}$$

Here, the orders \(s_1, s_2\) satisfy \(0<s_1 \ne s_2 <1\) (order of diffusion-type terms). We establish the local well-posedness of the solutions to the Cauchy problem when the source f is globally Lipschitz. In the case when the source term f satisfies a locally Lipschitz condition, the existence in large time, blow-up in finite time and continuous dependence on the initial data of the solutions are given.

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Acknowledgements

This research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2019.09

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam

    Nguyen Huy Tuan

  2. Vietnam National University, Ho Chi Minh City, Vietnam

    Nguyen Huy Tuan

  3. Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam

    Nguyen Huy Tuan

  4. Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh, 700000, Vietnam

    Vo Van Au

  5. Faculty of Natural Sciences, Duy Tan University, Da Nang, 550000, Vietnam

    Vo Van Au

  6. Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam

    Vo Viet Tri

  7. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

    Donal O’Regan

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  1. Nguyen Huy Tuan
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  2. Vo Van Au
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  3. Vo Viet Tri
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  4. Donal O’Regan
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Correspondence to Vo Viet Tri.

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Tuan, N.H., Au, V.V., Tri, V.V. et al. On the well-posedness of a nonlinear pseudo-parabolic equation. J. Fixed Point Theory Appl. 22, 77 (2020). https://doi.org/10.1007/s11784-020-00813-5

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  • DOI: https://doi.org/10.1007/s11784-020-00813-5

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