In this article, we consider the Cauchy problem for the following time-space fractional pseudo-parabolic equations
$ \begin{equation*} \left\{\begin{array}{l} { }_{0}^{C} D_{t}^{\alpha}(I-m \Delta ) u+\left ( - \Delta \right ) ^{\frac{\beta }{2} } u = |u|^{p-1} u, \quad x \in \mathbb{R}^{N}, \quad t>0, \\ u(0, x) = u_{0}(x), \quad\quad\quad\quad\quad\quad\quad\qquad x \in \mathbb{R}^{N}, \end{array} \right. \end{equation*} $
where $ 0 < \alpha < 1, \ 0 < \beta < 2, \ p > 1, \ m > 0, \ u_{0} \in L^{q}\left(\mathbb{R}^{N}\right) $. An estimating $ L^p-L^q $ for solution operator of time-space fractional pseudo-parabolic equations is obtained. The critical exponents of this problem are determined when $ u_0\in L^{q}(\mathbb{R}^{N}). $ Moreover, we also obtain global existence of the mild solution when $ u_0\in L^p(\mathbb{R}^{N})\cap L^q(\mathbb{R}^{N}) $ small enough.
Citation: Yaning Li, Yuting Yang. Blow-up and global existence of solutions for time-space fractional pseudo-parabolic equation[J]. AIMS Mathematics, 2023, 8(8): 17827-17859. doi: 10.3934/math.2023909
In this article, we consider the Cauchy problem for the following time-space fractional pseudo-parabolic equations
$ \begin{equation*} \left\{\begin{array}{l} { }_{0}^{C} D_{t}^{\alpha}(I-m \Delta ) u+\left ( - \Delta \right ) ^{\frac{\beta }{2} } u = |u|^{p-1} u, \quad x \in \mathbb{R}^{N}, \quad t>0, \\ u(0, x) = u_{0}(x), \quad\quad\quad\quad\quad\quad\quad\qquad x \in \mathbb{R}^{N}, \end{array} \right. \end{equation*} $
where $ 0 < \alpha < 1, \ 0 < \beta < 2, \ p > 1, \ m > 0, \ u_{0} \in L^{q}\left(\mathbb{R}^{N}\right) $. An estimating $ L^p-L^q $ for solution operator of time-space fractional pseudo-parabolic equations is obtained. The critical exponents of this problem are determined when $ u_0\in L^{q}(\mathbb{R}^{N}). $ Moreover, we also obtain global existence of the mild solution when $ u_0\in L^p(\mathbb{R}^{N})\cap L^q(\mathbb{R}^{N}) $ small enough.
| [1] |
B. de Andrade, A. Viana, Abstract Volterra integro-differential equations with applications to parabolic models with memory, Math. Ann., 369 (2017), 1131–1175. https://doi.org/10.1007/s00208-016-1469-z doi: 10.1007/s00208-016-1469-z
|
| [2] |
P. Biler, J. Dziubanski, W. Hebisch, Scattering of small solutions to generalized Benjamin-Bona-Mahony equation in several space dimensions, Commun. Partial Differ. Equations, 17 (1992), 1737–1758. https://doi.org/10.1080/03605309208820902 doi: 10.1080/03605309208820902
|
| [3] |
Y. Chen, H. Gao, M. J. Garrido-Atienza, B. Schmalfuss, Pathwise solutions of SPDEs driven by Holder-continuous integrators with exponent larger than $\frac{1}{2} $ and random dynamical systems, arXiv, 2014. https://doi.org/10.48550/arXiv.1305.6903 doi: 10.48550/arXiv.1305.6903
|
| [4] |
Y. Cao, J. Yin, C. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differ. Equations, 246 (2009), 4568–4590. https://doi.org/10.1016/j.jde.2009.03.021 doi: 10.1016/j.jde.2009.03.021
|
| [5] |
H. Dong, D. Kim, $L_p$-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math., 345 (2019), 289–345. https://doi.org/10.1016/j.aim.2019.01.016 doi: 10.1016/j.aim.2019.01.016
|
| [6] |
M. Fardi, M. Ghasemi, A numerical solution strategy based on error analysis for time-fractional mobile/immobile transport model, Soft Comput., 25 (2021), 11307–11331. https://doi.org/10.1007/s00500-021-05914-y doi: 10.1007/s00500-021-05914-y
|
| [7] |
M. Fardi, Y. Khan, A fast difference scheme on a graded mesh for time-fractional and space distributed-order diffusion equation with nonsmooth data, Int. J. Mod. Phys. B, 36 (2022), 15. https://doi.org/10.1142/S021797922250076X doi: 10.1142/S021797922250076X
|
| [8] |
M. Fardi, S. K. Q. Al-Omari, S. Araci, A pseudo-spectral method based on reproducing kernel for solving the time-fractional diffusion-wave equation, Adv. Contin. Discret Model., 2022 (2022), 54. https://doi.org/10.1186/s13662-022-03726-4 doi: 10.1186/s13662-022-03726-4
|
| [9] |
Y. Giga, T. Namba, Well-posedness of Hamilton-Jacobi equations with Caputo's time fractional derivative, Commun. Partial Differ. Equation, 42 (2017), 1088–1120. https://doi.org/10.1080/03605302.2017.1324880 doi: 10.1080/03605302.2017.1324880
|
| [10] | R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler functions, related topics and applications, Springer, 2014. |
| [11] | R. Hilfer, Applications of fractional calculus in physics, World Scientific, 2000. |
| [12] |
L. Jin, L. Li, S. Fang, The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation, Comput. Math. Appl., 73 (2017), 2221–2232. https://doi.org/10.1016/j.camwa.2017.03.005 doi: 10.1016/j.camwa.2017.03.005
|
| [13] | G. Karch, Asymptotic behaviour of solutions to some pesudoparabolic equations, Math. Methods Appl. Sci., 20 (1997), 271–289. |
| [14] |
M. Kirane, Y. Laskri, N. E. Tatar, Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives, J. Math. Anal. Appl., 312 (2005), 488–501. https://doi.org/10.1016/j.jmaa.2005.03.054 doi: 10.1016/j.jmaa.2005.03.054
|
| [15] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science, 2006. |
| [16] |
L. Li, J. G. Liu, L. Wang, Cauchy problems for Keller-Segel type time-space fractional diffusion equation, J. Differ. Equations, 265 (2018), 1044–1096. https://doi.org/10.1016/j.jde.2018.03.025 doi: 10.1016/j.jde.2018.03.025
|
| [17] |
Y. Li, Y. Yang, The critical exponents for a semilinear fractional pseudo-parabolic equation with nonlinear memory in a bounded domain, Electron. Res. Arch., 31 (2023), 2555–2567. https://doi.org/10.3934/era.2023129 doi: 10.3934/era.2023129
|
| [18] |
Y. Li, Q. Zhang, Blow-up and global existence of solutions for a time fractional diffusion equation, Fract. Calc. Appl. Anal., 21 (2018), 1619–1640. https://doi.org/10.1515/fca-2018-00859 doi: 10.1515/fca-2018-00859
|
| [19] | F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, Waves Stab. Contin. Media, 1994 (1994), 246–251. |
| [20] |
B. B. Mandelbrot, J. W. V. Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422–437. https://doi.org/10.1137/1010093 doi: 10.1137/1010093
|
| [21] |
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
|
| [22] |
E. Orsingher, L. Beghin, Fractional diffusion equations and processes with randomly varying time, Ann. Probab., 37 (2009), 206–249. https://doi.org/10.1214/08-AOP401 doi: 10.1214/08-AOP401
|
| [23] | I. Podlubny, Fractional differential euations, Elsevier Science, 1999. |
| [24] |
M. Ralf, K. Joseph, The random walk's 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
|
| [25] |
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
|
| [26] |
Y. F. Sun, Z. Zeng, J. Song, Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation, Numer. Algebra Control Optim., 10 (2020), 157–164. https://doi.org/10.3934/naco.2019045 doi: 10.3934/naco.2019045
|
| [27] |
N. H. Tuan, V. V. Au, R. Xu, Semilinear Caputo time-fractional pseudo-parabolic equations, Commun. Pure Appl. Anal., 20 (2020), 583–621. https://doi.org/10.3934/cpaa.2020282 doi: 10.3934/cpaa.2020282
|
| [28] |
K. Zennir, H. Dridi, S. Alodhaibi, S. Alkhalaf, Nonexistence of global solutions for coupled system of pseudoparabolic equations with variable exponents and weak memories, J. Funct. Space, 2021 (2021), 5573959. https://doi.org/10.1155/2021/5573959 doi: 10.1155/2021/5573959
|
| [29] |
K. Zennir, T. Miyasita, Lifespan of solutions for a class of pseudo-parabolic equation with weak-memory, Alexandria Eng. J., 59 (2020), 957–964. https://doi.org/10.1016/j.aej.2020.03.016 doi: 10.1016/j.aej.2020.03.016
|
| [30] |
Q. Zhang, H. Sun, The blow-up and global existence of solutions of Cauchy problems for a time fractional diffusion equation, Topol. Methods Nonlinear Anal., 46 (2015), 69–92. https://doi.org/10.12775/TMNA.2015.038 doi: 10.12775/TMNA.2015.038
|
Yaning Li, Yuting Yang. Blow-up and global existence of solutions for time-space fractional pseudo-parabolic equation[J]. AIMS Mathematics, 2023, 8(8): 17827-17859. doi: 10.3934/math.2023909
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