The Continuity by Order Derivative for Conformable Parabolic Equations with Exponential Nonlinearity

Binh Ho Duy; Dinh Huy Nguyen; Tien Nguyen Van; Dang Khoa Le

doi:10.61383/ejam.202531101

Authors

Ho Duy Binh Department of Mathematics, Faculty of Applied Science, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam; Vietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc City, Ho Chi Minh City, Vietnam; Nguyen Hue University - Second Army Academy, Dong Nai, Vietnam https://orcid.org/0000-0002-8827-5812
Nguyen Dinh Huy Department of Mathematics, Faculty of Applied Science, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam; Vietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc City, Ho Chi Minh City, Vietnam https://orcid.org/0000-0002-3755-0757
Nguyen Van Tien Department of Mathematics, FPT University, Ha Noi, Vietnam Corresponding Author https://orcid.org/0000-0002-0975-9131
Le Dang Khoa Gia Dinh High school, Ho Chi Minh City, Vietnam https://orcid.org/0009-0001-0189-5796

DOI:

https://doi.org/10.61383/ejam.202531101

Keywords:

initial value problem, Sobolev embeddings, parabolic equations, exponential non-linearity, conformable derivative

Abstract

In this article, we examine the continuity according to the derivative order of conformable parabolic equations. We establish the existence and uniqueness of mild solutions to the problem of source function exponential non-linearity. We prove the existence and uniqueness of a mild solution based on some Sobolev embeddings, the Banach space of all continuous functions, and the Banach fixed point theorem. In addition, we will address the nonlinear model's continuity issue and demonstrate the mild solution's convergence to the nonlinear problem when \(\gamma'\to \gamma\).

References

[1] A. Alsaedi, B. Ahmad, M. Kirane, and B. T. Torebek, Blowing-up solutions of the time-fractional dispersive equations, Adv. Nonlinear Anal. 10 (2021), no. 1, 952–971. DOI: https://doi.org/10.1515/anona-2020-0153

[2] P. M. de Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in ℕ^R, J. Differential Equations 259 (2015), no. 7, 2948–2980. DOI: https://doi.org/10.1016/j.jde.2015.04.008

[3] R. Grande, Space-Time Fractional Nonlinear Schrödinger Equation, SIAM J. Math. Anal. 51 (2019), no. 5, 4172–4212. DOI: https://doi.org/10.1137/19M1247140

[4] M. Ishiwata, B. Ruf, F. Sani, and E. Terraneo, Asymptotics for a parabolic equation with critical exponential nonlinearity, J. Evol. Equ. (2020), 1–40. DOI: https://doi.org/10.1007/s00028-020-00649-z

[5] M. Ruzhansky, N. Tokmagambetov, and B. T. Torebek, On a non-local problem for a multi-term fractional diffusion-wave equation, Fract. Calc. Appl. Anal. 23 (2020), no. 2, 324–355. DOI: https://doi.org/10.1515/fca-2020-0016

[6] M. Ruzhansky, D. Serikbaev, B. T. Torebek, and N. Tokmagambetov, Direct and inverse problems for time-fractional pseudo-parabolic equations, Quaest. Math. (2021), 1–19. DOI: https://doi.org/10.2989/16073606.2021.1928321

[7] N. H. Tuan, V. V. Au, and R. Xu, Semilinear Caputo time-fractional pseudo-parabolic equations, Commun. Pure Appl. Anal. 20 (2021), no. 2, 583. DOI: https://doi.org/10.3934/cpaa.2020282

[8] N. H. Tuan, Y. Zhou, T. N. Thach, and N. H. Can, Initial inverse problem for the nonlinear fractional Rayleigh-Stokes equation with random discrete data, Commun. Nonlinear Sci. Numer. Simul. 78 (2019), 104873. DOI: https://doi.org/10.1016/j.cnsns.2019.104873

[9] T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A 272 (1972), no. 1220, 47–78. DOI: https://doi.org/10.1098/rsta.1972.0032

[10] W. T. Ting, Certain non-steady flows of second-order fluids, Arch. Ration. Mech. Anal. 14 (1963), no. 1, 1–26. DOI: https://doi.org/10.1007/BF00250690

[11] V. Padron, Effect of aggregation on population recovery modeled by a forward-backward pseudo-parabolic equation, Trans. Amer. Math. Soc. 356 (2004), no. 7, 2739–2756. DOI: https://doi.org/10.1090/S0002-9947-03-03340-3

[12] H. D. Binh, L. N. Hoang, D. Baleanu, and H. T. K. Van, Continuity result on the order of a nonlinear fractional pseudo-parabolic equation with Caputo derivative, Fractal Fract. 5 (2021), no. 2, 41. DOI: https://doi.org/10.3390/fractalfract5020041

[13] E. Karapinar, H. D. Binh, N. H. Luc, and N. H. Can, On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems, Adv. Differ. Equ. 2021 (2021), no. 1, 1–24. DOI: https://doi.org/10.1186/s13662-021-03232-z

[14] R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65–70. DOI: https://doi.org/10.1016/j.cam.2014.01.002

[15] G. Furioli, T. Kawakami, B. Ruf, and E. Terraneo, Asymptotic behavior and decay estimates of the solutions for a nonlinear parabolic equation with exponential nonlinearity, J. Differential Equations 262 (2017), no. 1, 145–180. DOI: https://doi.org/10.1016/j.jde.2016.09.024

[16] A. Z. Fino and M. Kirane, The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity, Commun. Pure Appl. Anal. 19 (2020), no. 7, 3625–3650. DOI: https://doi.org/10.3934/cpaa.2020160

[17] N. Ioku, The Cauchy problem for heat equations with exponential nonlinearity, J. Differential Equations 251 (2011), no. 4–5, 1172–1194. DOI: https://doi.org/10.1016/j.jde.2011.02.015

[18] M. Majdoub, S. Otsmane, and S. Tayachi, Local well-posedness and global existence for the biharmonic heat equation with exponential nonlinearity, Adv. Differential Equations 23 (2018), no. 7–8, 489–522. DOI: https://doi.org/10.57262/ade/1526004064

[19] M. Majdoub and S. Tayachi, Global existence and decay estimates for the heat equation with exponential nonlinearity, Funkcial. Ekvac. 64 (2021), no. 2, 237–259. DOI: https://doi.org/10.1619/fesi.64.237

[20] M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order, J. Funct. Anal. 155 (1998), no. 2, 364–380. DOI: https://doi.org/10.1006/jfan.1997.3236

[21] N. H. Luc, D. Lan, D. O’Regan, N. A. Tuan, and Y. Zhou, On the initial value problem for the nonlinear fractional Rayleigh-Stokes equation, J. Fixed Point Theory Appl. 23 (2021), no. 4, 1–28. DOI: https://doi.org/10.1007/s11784-021-00897-7

[22] D. Lan, Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations, Evol. Equ. Control Theory 11 (2022), no. 1. DOI: https://doi.org/10.3934/eect.2021002

[23] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), 426–447. DOI: https://doi.org/10.1016/j.jmaa.2011.04.058

[24] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer-Verlag, New York, 2011. DOI: https://doi.org/10.1007/978-0-387-70914-7

[25] A. Jaiswal and D. Bahuguna, Semilinear conformable fractional differential equations in Banach spaces, Differ. Equ. Dyn. Syst. 27 (2019), no. 1, 313–325. DOI: https://doi.org/10.1007/s12591-018-0426-6

[26] M. Li, J. R. Wang, and D. O’Regan, Existence and Ulam’s stability for conformable fractional differential equations with constant coefficients, Bull. Malays. Math. Sci. Soc. 42 (2019), no. 4, 1791–1812. DOI: https://doi.org/10.1007/s40840-017-0576-7

[27] N. H. Tuan, D. O’Regan, and T. B. Ngoc, Continuity with respect to fractional order of the time fractional diffusion-wave equation, Evol. Equ. Control Theory 9 (2020), no. 3, 773–793. DOI: https://doi.org/10.3934/eect.2020033