On maximal solution to a degenerate parabolic equation involving in time fractional derivative
Authors
-
Bui Dai Nghia
Department of Mathematics, Faculty of Science, Nong Lam University, Ho Chi Minh City, Vietnam
https://orcid.org/0009-0003-8545-0683
-
Nguyen Hoang Luc
Department of Mathematical Economics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam
https://orcid.org/0000-0001-9664-6743
- Xiaolan Qin School of Mathematical Science, Guizhou Normal University, Guiyang 550001, China
- Yan Wang School of Mathematical Science, Guizhou Normal University, Guiyang 550001, China
https://doi.org/10.61383/ejam.20231129
Keywords:
degenerate parabolic equation, fractional integrodifferential equation, mild solution, maximal solutionAbstract
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
M.M. El-Borai Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fractals 14 (2002) 433–440. DOI: https://doi.org/10.1016/S0960-0779(01)00208-9
R. Hilfer, Fractional calculus in Physics, World Scientific, Singapore, 2000. DOI: https://doi.org/10.1142/3779
F. Li, J. Liang, 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
J. Liang, T.J. Xiao, Abstract degenerate Cauchy problems in locally convex spaces, J. Math. Anal. Appl. 259 (2001) 398–412. DOI: https://doi.org/10.1006/jmaa.2000.7406
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. DOI: https://doi.org/10.1007/978-1-4612-5561-1
I. Podlubny, Fractional differential equations, Academic Press, London, 1999.
Stefan G. Samko, Anatoly A. Kilbas and Oleg I. Marichev, Fractional integrals and derivatives,
Theory and Applications, Gordon and Breach Science, Naukai Tekhnika, Minsk (1987).
Tuan, Nguyen Huy; Caraballo, Tomás; On initial and terminal value problems for fractional nonclassical diffusion equations Proc. Amer. Math. Soc. 149 (2021), no. 1, 143–161. DOI: https://doi.org/10.1090/proc/15131
A. Lunardi, Analyticity of the maximal solution of an abstract nonlinear parabolic equation Nonlinear Anal. 6 (1982), no. 5, 503--521. DOI: https://doi.org/10.1016/0362-546X(82)90062-1
A. Meraj, 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
A.T. Nguyen, T. Caraballo, N.H. Tuan, On the initial value problem for a class of nonlinear biharmonic equation with time-fractional derivative Proc. Roy. Soc. Edinburgh Sect. A 152 (2022), no. 4, 989–1031. DOI: https://doi.org/10.1017/prm.2021.44
T. Caraballo, T.B. Ngoc, N.H. Tuan, R. Wang, On a nonlinear Volterra integrodifferential equation involving fractional derivative with Mittag-Leffler kernel Proc. Amer. Math. Soc. 149 (2021), no. 08, 3317-3334. DOI: https://doi.org/10.1090/proc/15472
T.B. Ngoc, T. Caraballo, N.H. Tuan, 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
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Bui Dai Nghia; Nguyen Hoang Luc (Corresponding Author); Xiaolan Qin, Yan Wang
This work is licensed under a Creative Commons Attribution 4.0 International License.
Accepted 2023 May 04
Published 2023 May 08
