Blow-up of solutions for a time fractional biharmonic equation with exponentional nonlinear memory

Yuchen Zhu; Yuchen Zhu

doi:10.3934/era.2024278

Electronic Research Archive

2024, Volume 32, Issue 11: 5988-6007. doi: 10.3934/era.2024278

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Research article

Blow-up of solutions for a time fractional biharmonic equation with exponentional nonlinear memory

Yuchen Zhu ^,

College of Mathematics & Statistics, Nanjing University of Information Science & Technology, Nanjing 210044, China

Received: 30 September 2024 Revised: 29 October 2024 Accepted: 06 November 2024 Published: 08 November 2024

In the paper, we focus on the local existence and blow-up of solutions for a time fractional nonlinear equation with biharmonic operator and exponentional nonlinear memory in an Orlicz space. We first establish a $ L^p-L^q $ estimate for solution operators of a time fractional nonlinear biharmonic equation, and obtain bilinear estimates for mild solutions. Then, based on the contraction mapping principle, we establish the local existence of mild solutions. Moreover, by using the test function method, we obtain the blow-up result of solutions.
- fractional biharmonic equation,
- exponentional nonlinear memory,
- blow-up,
- local existence
Citation: Yuchen Zhu. Blow-up of solutions for a time fractional biharmonic equation with exponentional nonlinear memory[J]. Electronic Research Archive, 2024, 32(11): 5988-6007. doi: 10.3934/era.2024278

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Abstract

In the paper, we focus on the local existence and blow-up of solutions for a time fractional nonlinear equation with biharmonic operator and exponentional nonlinear memory in an Orlicz space. We first establish a $ L^p-L^q $ estimate for solution operators of a time fractional nonlinear biharmonic equation, and obtain bilinear estimates for mild solutions. Then, based on the contraction mapping principle, we establish the local existence of mild solutions. Moreover, by using the test function method, we obtain the blow-up result of solutions.

References

[1]	A. A. Kilbas, H. M. Srivastava, J. J. Trugillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
[2]	R. Metzler, J. Klafter, The random walks guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1–77. https://doi.org/10.1016/s0370-1573(00)00070-3 doi: 10.1016/s0370-1573(00)00070-3
[3]	R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), 161–208. https://doi.org/10.1088/0305-4470/37/31/r01 doi: 10.1088/0305-4470/37/31/r01
[4]	R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Status (B): Basic Res., 133 (1986), 425–430. https://doi.org/10.1002/pssb.2221330150 doi: 10.1002/pssb.2221330150
[5]	W. R. Schneider, W. Wyss, Fractional diffusion and wave equations, J. Math. Phys., 30 (1989), 134–144. https://doi.org/10.1063/1.528578 doi: 10.1063/1.528578
[6]	R. Wang, D. Chen, T. Xiao, Abstract fractional cauchy problems with almost sectorial operators, J. Differ. Equations, 252 (2012), 202–235. https://doi.org/10.1016/j.jde.2011.08.048 doi: 10.1016/j.jde.2011.08.048
[7]	Z. Zhang, D. Wang, J. Chen, Z. Xie, C. Xu, Blow-up of solutions for the fourth-order schrödinger equation with combined power-type nonlinearities, J. Geom. Anal., 34 (2024), 1–36. https://doi.org/10.1007/s12220-024-01747-x doi: 10.1007/s12220-024-01747-x
[8]	Z. Zhang, Z. Liu, Y. Deng, L. Li, F. He, C. Huang, A trilinear estimate with application to the perturbed nonlinear Schrödinger equations with the Kerr law nonlinearity, J. Evol. Equations, 21 (2021), 1–18. https://doi.org/10.1007/s00028-020-00631-9 doi: 10.1007/s00028-020-00631-9
[9]	Z. Zhang, Z. Liu, M. Sun, S. Li, Low regularity for the higher order nonlinear dispersive equation in Sobolev spaces of negative index, J. Dyn. Differ. Equations, 31 (2019), 419–433. https://doi.org/10.1007/s10884-018-9669-8 doi: 10.1007/s10884-018-9669-8
[10]	Z. Zhang, Z. Liu, Y. Deng, J. Huang, C. Huang, Long time behavior of solutions to the damped forced generalized Ostrovsky equation below the energy space, P. Am. Math. Soc., 149 (2021), 1527–1542. https://doi.org/10.1090/proc/15322 doi: 10.1090/proc/15322
[11]	Z. Zhang, Z. Liu, M. Sun, S. Li, Well-posedness and unique continuation property for the solutions to the generalized Kawahara equation below the energy space, Appl. Anal., 97 (2018), 2655–2685. https://doi.org/10.1080/00036811.2017.1385064 doi: 10.1080/00036811.2017.1385064
[12]	N. Ioku, The Cauchy problem for heat equations with exponential nonlinearity, J. Differ. Equations, 251 (2011), 1172–1194. https://doi.org/10.1016/j.jde.2011.02.015 doi: 10.1016/j.jde.2011.02.015
[13]	N. Ioku, B. Ruf, E. Terraneo, Existence, non-existence, and uniqueness for a heat equation with exponential nonlinearity in $\mathbb{R}^2$, Math. Phys. Anal. Geom., 18 (2015), 1–19. https://doi.org/10.1007/s11040-015-9199-0 doi: 10.1007/s11040-015-9199-0
[14]	G. Furioli, T. Kawakami, B. Ruf, E. Terraneo, Asymptotic behavior and decay estimates of the solutions for a nonlinear parabolic equation with exponential nonlinearity, J. Differ. Equations, 262 (2017), 145–180. https://doi.org/10.1016/j.jde.2016.09.024 doi: 10.1016/j.jde.2016.09.024
[15]	M. Majdoub, S.Otsmane, S. Tayachi, Local well-posedness and global existence for the biharmonic heat equation with exponential nonlinearity, Adv. Differ. Equations, 23 (2018), 489–522. https://doi.org/10.57262/ade/1526004064 doi: 10.57262/ade/1526004064
[16]	M. Majdoub, S. Tayachi, Well-posedness, Global existence and decay estimates for the heat equation with general power-exponential nonlinearities, Proc. Int. Cong. Math., 4 (2018), 2413–2438. https://doi.org/10.1142/9789813272880_0144 doi: 10.1142/9789813272880_0144
[17]	M. Majdoub, S. Tayachi, Global existence and decay estimates for the heat equation with exponential nonlinearity, Funkcialaj Ekvacioj, 64 (2021), 237–259. https://doi.org/10.1619/fesi.64.237 doi: 10.1619/fesi.64.237
[18]	A. Z. Fino, M. Kirane, The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity, Commun. Pur. Appl. Anal., 19 (2020), 3625–3650. https://doi.org/10.3934/cpaa.2020160 doi: 10.3934/cpaa.2020160
[19]	N. A. Tuan, T. Caraballo, N. H. Tuan, On the initial value problem for a class of nonlinear biharmonic equation with time-fractional derivative, P. Roy. Soc. Edinb. A, 152 (2021), 989–1031. https://doi.org/10.1017/prm.2021.44 doi: 10.1017/prm.2021.44
[20]	N. A. Tuan, C. Yang, On a time-space fractional diffusion equation with a semilinear source of exponential type, Electron. Res. Arch., 30 (2022), 1354–1373. https://doi.org/10.3934/era.2022071 doi: 10.3934/era.2022071
[21]	J. W. Cholewa, A. Rodriguez-Berna, Linear and semilinear higher order parabolic equations in $\mathbb{R}^n$, Nonlinear Anal. Theor., 75 (2012), 194–210. https://doi.org/10.1016/j.na.2011.08.022 doi: 10.1016/j.na.2011.08.022
[22]	R. Wang, N. H. Can, A. T. Nguyen, N. H. Tuan, Local and global existence of solutions to a time-fractional wave equation with an exponential growth, Commun. Nonlinear Sci., 118 (2023). https://doi.org/10.1016/j.cnsns.2022.107050 doi: 10.1016/j.cnsns.2022.107050
[23]	A. Z. Fino, M. Kirane, Qualitative properties of solutions to a time-space fractional evolution equation, Q. Appl. Math., 70 (2012), 133–157. https://doi.org/10.1090/s0033-569x-2011-01246-9 doi: 10.1090/s0033-569x-2011-01246-9
[24]	B. Ahmad, A. Alsaedi, M. Kirane, On a reaction diffusion equation with nonlinear time-nonlocal source term, Math. Meth. Appl. Sci., 39 (2016), 236–244. https://doi.org/10.1002/mma.3473 doi: 10.1002/mma.3473
[25]	A. Bekkai, B. Rebiai, M. Kirane, On local existence and blowup of solutions for a time-space fractional diffusion equation with exponential nonlinearity, Math. Meth. Appl. Sci., 42 (2019), 1819–1830. https://doi.org/10.1002/mma.5476 doi: 10.1002/mma.5476
[26]	R. A. Adams, J. J. F. Fourier, Sobolev Spaces, Elsevier, 2003.
[27]	M. M. Rao, Z. D. Ren, Applications of Orlicz Spaces, CRC Press, 2002.
[28]	I. Podlubny, Fractional differential equations, Math. Sci. Eng., 198 (1999), 41–119.
[29]	Q. Zhang, Y. Li, Global well-posedness and blow-up solutions of the Cauchy problem for a time-fractional superdiffusion equation, J. Evol. Equations, 19 (2019), 271–303. https://doi.org/10.1007/s00028-018-0475-x doi: 10.1007/s00028-018-0475-x
[30]	J. D. Djida, A. Fernandez, I. Area, Well-posedness results for fractional semi-linear wave equations, Discrete Contin. Dyn. Syst.-Ser. B, 25 (2020), 569–597. https://doi.org/10.3934/dcdsb.2019255 doi: 10.3934/dcdsb.2019255
[31]	A. Z. Fino, Blow-up rates for a higher-order semilinear parabolic equation with nonlinear memory term, Appl. Anal., 101 (2022), 4775–4792. https://doi.org/10.1080/00036811.2020.1869945 doi: 10.1080/00036811.2020.1869945

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