Blow-up solutions to the fractional solid fuel ignition model

Authors

Nguyen Dinh Huy Department of Mathematics, Faculty of Applied Science, Ho Chi Minh City University of Technology (HCMUT), 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Vietnam https://orcid.org/0000-0002-3755-0757
Anh Tuan Nguyen Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam Corresponding Author https://orcid.org/0000-0002-8757-9742

DOI:

https://doi.org/10.61383/ejam.20231126

Keywords:

Caputo-Fabrizio, fractional equation, exponential nonlinearity

Abstract

In this study, we investigate a time fractional parabolic equation with Caputo-Fabrizio derivative and a pure exponential source term. The model can be seen as a modified version of a solid fuel ignition by considering delay effects. Because of the bad behavior of the source term, the common Banach fixed point theorem is not suitable to be applied. Therefore, we approach by an iteration method in which we find a supersolution to the problem. Then, the monotony of approximating solutions implies the existence of a mild solution. Furthermore, we show that if the given initial data is sufficiently large in term of norm measure, the corresponding solution will blow up in a finite time. Main techniques of the work are mainly based on calculations related to explicit formulas of solution operator derived from eigenpair of Helmholtz's equation and Sobolev embeddings between Hilbert scale spaces.

References

N.H. Tuan, and Y. Zhou, Well-posedness of an initial value problem for fractional diffusion equation with Caputo-Fabrizio derivative, Journal of Computational and Applied Mathematics 375 , 112811, (2020), pp 1-21, DOI: https://doi.org/10.1016/j.cam.2020.112811 DOI: https://doi.org/10.1016/j.cam.2020.112811

X. Zheng, H. Wang, and H. Fu, Well-posedness of fractional differential equations with variable-order Caputo-Fabrizio derivative, Chaos Solitons Fractals 138, no. 109966, (2020) ,pp. 1-7, DOI: https://doi.org/10.1016/j.chaos.2020.109966 DOI: https://doi.org/10.1016/j.chaos.2020.109966

N.H. Tuan, Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation, Discrete and Continuous Dynamical Systems - S 14, no. 12, (2021) , pp. 4551-4574, DOI: 10.3934/dcdss.2021113 DOI: https://doi.org/10.3934/dcdss.2021113

N.H. Tuan, A.T. Nguyen, D. O'Regan, and V.V. Tri, On the initial value problem for fractional Volterra integrodifferential equations with a Caputo-Fabrizio derivative, Mathematical Modelling of Natural Phenomena 16, no. 18, (2021), pp. 1-21, DOI: https://doi.org/10.1051/mmnp/2021010 DOI: https://doi.org/10.1051/mmnp/2021010

N.H. Luc, J. Singh, N.P.Q. Trang, and H.T.K. Van, On inverse problem for linear and semilinear-diffusion equation with Caputo–Fabrizio derivative, Mathematical Methods in Applied Science, (2021), DOI: https://doi.org/10.1002/mma.7766. DOI: https://doi.org/10.1002/mma.7766

L. Evans, Partial differential equations, Vol. 19, American Mathematical Society, 2022

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Published

2023 May 07

Issue

Vol. 1 No. 1 (2023)

Section

Research Article

License

This work is licensed under a Creative Commons Attribution 4.0 International License.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

How to Cite

[1]

“ Blow-up solutions to the fractional solid fuel ignition model”, Electron. J. Appl. Math., vol. 1, no. 1, pp. 30–39, May 2023, doi: 10.61383/ejam.20231126.

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