On maximal solution to a degenerate parabolic equation involving a time fractional derivative

Authors

  • Bui Dai Nghia Department of Mathematics, Faculty of Science, Nong Lam University, Ho Chi Minh City, Vietnam
  • Nguyen Hoang Luc Department of Mathematical Economics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam Corresponding Author
  • Xiaolan Qin School of Mathematical Science, Guizhou Normal University, Guiyang 550001, China
  • Yan Wang School of Mathematical Science, Guizhou Normal University, Guiyang 550001, China

DOI:

https://doi.org/10.61383/ejam.20231129

Keywords:

degenerate parabolic equation, fractional integrodifferential equation, mild solution, maximal solution

Abstract

In this paper, we consider a degenerate parabolic equation associated with Caputo derivative. Our problem is studied in the unbounded domain and the nonlocal initial condition. Under some suitable conditions of the input data, we show the local existence of the mild solution. Then we show the continuation of the mild solution. We also claim the maximal mild solution. The main analysis in the current paper is based on some estimations of resolvent theory combined with many complex valuations on solutions operators in Banach spaces.

References

[1] Tom´as Caraballo, T. B. Ngoc, N. H. Tuan, and R. Wang, On a nonlinear volterra integrodifferential equation involving fractional derivative with mittag-leffler kernel, Proceedings of the American Mathematical Society 149 (2021), no. 8, 3317–3334. DOI: https://doi.org/10.1090/proc/15472

[2] Nguyen Duc Phuong, Dumitru Baleanu, Tran Thanh Phong, and Le Dinh Long, Recovering the source term for parabolic equation with nonlocal integral condition, Mathematical Methods in the Applied Sciences 44 (2021), no. 11, 9026–9041, ISSN 1099-1476, DOI 10.1002/mma.7331, http://dx.doi.org/10.1002/mma.7331. DOI: https://doi.org/10.1002/mma.7331

[3] F. Li, J. Liang, and H.-K. Xu, Existence of mild solutions for fractional integrodifferential equations of sobolev type with nonlocal conditions, Journal of Mathematical Analysis and Applications (2012), 510–525. DOI: https://doi.org/10.1016/j.jmaa.2012.02.057

[4] J. Liang and T. J. Xiao, Abstract degenerate cauchy problems in locally convex spaces, Journal of Mathematical Analysis and Applications 259 (2001), 398–412. DOI: https://doi.org/10.1006/jmaa.2000.7406

[5] Le Dinh Long, Nguyen Hoang Luc, and Nguyen Duc Phuong, Identifying inverse source for diffusion equation with conformable time derivative by fractional Tikhonov method, Advances in the Theory of Nonlinear Analysis and its Application 6 (2022), no. 4, 433–450. DOI: https://doi.org/10.31197/atnaa.1079951

[6] A. Meraj and N. D. Pandey, Existence and uniqueness of extremal mild solutions for non-autonomous nonlocal integro-differential equations via monotone iterative technique, Filomat 33 (2019), no. 10, 2985–2993. DOI: https://doi.org/10.2298/FIL1910985M

[7] T. B. Ngoc, Tom´as Caraballo, N. H. Tuan, and Y. Zhou, Existence and regularity results for terminal value problem for nonlinear fractional wave equations, Nonlinearity 34 (2021), no. 3, 1448–1503. DOI: https://doi.org/10.1088/1361-6544/abc4d9

[8] A. T. Nguyen, Tom´as Caraballo, and N. H. Tuan, On the initial value problem for a class of nonlinear biharmonic equation with time-fractional derivative, Proceedings of the Royal Society of Edinburgh: Section A Mathematics 152 (2022), no. 4, 989–1031. DOI: https://doi.org/10.1017/prm.2021.44

[9] Huy Tuan Nguyen and Tom´as Caraballo, On initial and terminal value problems for fractional nonclassical diffusion equations, Proceedings of the American Mathematical Society 149 (2021), no. 1, 143–161. DOI: https://doi.org/10.1090/proc/15131

[10] Nguyen Duc Phuong, Luu Vu Cam Hoan, Erdal Karapinar, Jagdev Singh, Ho Duy Binh, and Nguyen Huu Can, Fractional order continuity of a time semi-linear fractional diffusion-wave system, Alexandria Engineering Journal 59 (2020), no. 6, 4959–4968, ISSN 1110-0168, DOI 10.1016/j.aej.2020.08.054, http://dx.doi.org/10.1016/j.aej.2020.08.054. DOI: https://doi.org/10.1016/j.aej.2020.08.054

[11] Nguyen Duc Phuong, Le Dinh Long, Devender Kumar, and Ho Duy Binh, Determine unknown source problem for time fractional pseudo-parabolic equation with atangana-baleanu caputo fractional derivative, AIMS Mathematics 7 (2022), no. 9, 16147–16170, ISSN 2473-6988, DOI 10.3934/math.2022883, http://dx.doi.org/10.3934/math.2022883. DOI: https://doi.org/10.3934/math.2022883

[12] Nguyen Duc Phuong, Le Dinh Long, Anh Tuan Nguyen, and Dumitru Baleanu, Regularization of the inverse problem for time fractional pseudo-parabolic equation with non-local in time conditions, Acta Mathematica Sinica, English Series 38 (2022), no. 12, 2199–2219, ISSN 1439-7617, DOI 10.1007/s10114-022-1234-z, http://dx.doi.org/10.1007/s10114-022-1234-z. DOI: https://doi.org/10.1007/s10114-022-1234-z

[13] Nguyen Duc Phuong, Van Tien Nguyen, and Le Dinh Long, Inverse source problem for sobolev equation with fractional laplacian, Journal of Function Spaces 2022 (2022), 1–12, ISSN 2314-8896, DOI 10.1155/2022/1035118, http://dx.doi.org/10.1155/2022/1035118. DOI: https://doi.org/10.1155/2022/1035118

[14] Nguyen Duc Phuong, Nguyen Huy Tuan, Zakia Hammouch, and Rathinasamy Sakthivel, On a pseudo-parabolic equations with a non-local term of the kirchhoff type with random gaussian white noise, Chaos, Solitons and Fractals 145 (2021), 110771, ISSN 0960-0779, DOI 10.1016/j.chaos.2021.110771, http://dx.doi.org/10.1016/j.chaos.2021.110771. DOI: https://doi.org/10.1016/j.chaos.2021.110771

[15] Tran Ngoc Thach, Devendra Kumar, Nguyen Hoang Luc, and Nguyen Duc Phuong, On a semilinear fractional reaction-diffusion equation with nonlocal conditions, Alexandria Engineering Journal 60 (2021), no. 6, 5511–5520, ISSN 1110-0168, DOI 10.1016/j.aej.2021.04.005, http://dx.doi.org/10.1016/j.aej.2021.04.005. DOI: https://doi.org/10.1016/j.aej.2021.04.005

[16] Nguyen Hoang Tuan, Nguyen Anh Triet, Nguyen Hoang Luc, and Nguyen Duc Phuong, On a time fractional diffusion with nonlocal in time conditions, Advances in Difference Equations 2021 (2021), no. 1, ISSN 1687-1847, DOI 10.1186/s13662-021-03365-1, http://dx.doi.org/10.1186/s13662-021-03365-1. DOI: https://doi.org/10.1186/s13662-021-03365-1

[17] Vo Van Au, Nguyen Duc Phuong, Nguyen Huy Tuan, and Yong Zhou, Some regularization methods for a class of nonlinear fractional evolution equations, Computers and Mathematics with Applications 78 (2019), no. 5, 1752–1771, ISSN 0898-1221, DOI 10.1016/j.camwa.2019.06.015, http://dx.doi.org/10.1016/j.camwa.2019.06.015. DOI: https://doi.org/10.1016/j.camwa.2019.06.015

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Published

2023 May 08

Issue

Vol. 1 No. 1 (2023)

Section

Research Article

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Copyright (c) 2023 Bui Dai Nghia; Nguyen Hoang Luc (Corresponding Author); Xiaolan Qin, Yan Wang

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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]
“On maximal solution to a degenerate parabolic equation involving a time fractional derivative”, Electron. J. Appl. Math., vol. 1, no. 1, pp. 62–80, May 2023, doi: 10.61383/ejam.20231129.

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