Abstract
In a Hilbert space, we consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable t and the space fractional function of the self-adjoint positive unbounded operator. We consider various cases of global Lipschitz and local Lipschitz source with time-singular coefficient. These sources are generalized of the well–known fractional equations such as the fractional Cahn–Allen equation, the fractional Burger equation, the fractional Cahn–Hilliard equation, the fractional Kuramoto–Sivashinsky equation, etc. Under suitable assumptions, we investigate the existence, uniqueness of maximal solution, and stability of solution of the problems with respect to perturbed fractional orders. We also establish some global existence and prove that the global solution can be approximated by known asymptotic functions as \(t\rightarrow \infty \).
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Acknowledgements
The authors thank an anonymous referee and the editor for helpful comments that improved the quality and presentation of the paper. The research was supported by Vietnam National University of Hochiminh City [Grant No. B2021-18-02]. The research of Erkan Nane was partially supported by the Simons Foundation Collaboration Grants for Mathematicians.
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Dien, N.M., Nane, E., Minh, N.D. et al. Global solutions of nonlinear fractional diffusion equations with time-singular sources and perturbed orders. Fract Calc Appl Anal 25, 1166–1198 (2022). https://doi.org/10.1007/s13540-022-00056-w
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DOI: https://doi.org/10.1007/s13540-022-00056-w
Keywords
- Fractional diffusion (primary)
- Caputo derivative
- Initial value problem
- Maximal solution
- Global solution
- Decay rate