Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations

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March 2022, 27(3): 1301-1322. doi: 10.3934/dcdsb.2021091

Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations

Yongqiang Fu and Xiaoju Zhang ^,

School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

^* Corresponding author: Xiaoju Zhang

Received October 2020 Published March 2022 Early access March 2021

Fund Project: The first author is supported by the National Natural Science Foundation of China grant No.11771107

In this paper, we investigate initial boundary value problems for Kirchhoff-type diffusion equations $ \partial_{t}^{\beta}u+M(\|u\|_{H_0^{s}(\Omega)}^2)(-\Delta)^{s}u = \gamma|u|^{\rho}u+g(t,x) $ with the Caputo time fractional derivatives and fractional Laplacian operators. We establish a new compactness theorem concerning time fractional derivatives. By Galerkin method, let $ 0<\rho<\frac{4s}{N-2s} $ when $ \gamma<0 $, and $ 0<\rho<\min\{\frac{4s}{N},\frac{2s}{N-2s}\} $ when $ \gamma>0 $, then we obtain the global existence and uniqueness of weak solutions for Kirchhoff problems. Furthermore, we get the decay properties of weak solutions in $ L^2(\Omega) $ and $ L^{\rho+2}(\Omega) $. Remarkably, the decay rate differs from that in the case $ \beta = 1 $.

Keywords: Kirchhoff-type diffusion, Time-space fractional, Global existence, Decay property, Weak solution.

Mathematics Subject Classification: Primary: 35R11; Secondary: 35A01, 35B40.

Citation: Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1301-1322. doi: 10.3934/dcdsb.2021091

References:

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[18]	L. Li, J. G. Liu and L. Z. Wang, Cauchy problems for Keller-Seqel type time-space fractional diffusion equation, J. Differential Equations, 265 (2018), 1044-1096. doi: 10.1016/j.jde.2018.03.025.
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[20]	P. Pucci, M. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional $p$-Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806. doi: 10.1007/s00526-015-0883-5.
[21]	P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22. doi: 10.4171/RMI/879.
[22]	N. Pan, B. L. Zhang and J. Cao, Degenerate Kirchhoff-type diffusion problems involving the fractional $p$-Laplacian, Nonlinear Anal. Real World Appl., 37 (2017), 56-70. doi: 10.1016/j.nonrwa.2017.02.004.
[23]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.
[24]	R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2015-2137. doi: 10.3934/dcds.2013.33.2105.
[25]	V. Vergara and R. Zacher, Lyapunov functions and convergence to steady state for differential equations of fractional order, Math. Z., 259 (2008), 287-309. doi: 10.1007/s00209-007-0225-1.
[26]	V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods,, SIAM J. Math. Anal., 47 (2015), 210-239. doi: 10.1137/130941900.
[27]	M. Q. Xiang and Y. Q. Fu, Weak solutions for nonlocal evolutional inequalities involving gradient constraints and variable exponent, Electron. J. Differential Equations, 2013 (2013), 1-17. doi: 10.1186/1687-2770-2013-96.
[28]	M. Q. Xiang, V. D. Rǎdulescu and B. L. Zhang, Nonlocal Kirchhoff diffusion problems: Local existence and blow-up of solutions, Nonlinearity, 31 (2018), 3228-3250. doi: 10.1088/1361-6544/aaba35.
[29]	M. Q. Xiang, B. L. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional $p$-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041. doi: 10.1016/j.jmaa.2014.11.055.
[30]	R. Zacher, Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients, J. Math. Anal. Appl., 348 (2008), 137-149. doi: 10.1016/j.jmaa.2008.06.054.
[31]	R. Zacher, Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces, Funkc. Ekvac., 52 (2009), 1-18. doi: 10.1619/fesi.52.1.
[32]	R. Zacher, A De Giorgi-Nash type theorem for time fractional diffusion equations,, Math. Ann., 356 (2013), 99-146. doi: 10.1007/s00208-012-0834-9.
[33]	Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations,, Comput. Math. Appl., 59 (2010), 1063-1077. doi: 10.1016/j.camwa.2009.06.026.
[34]	Q. G. Zhang, H. R. Sun and Y. N. Li, Global existence and blow-up of solutions of Cauchy problems for a time fractional diffusion system, Comput. Math. Appl., 78 (2019), 1357-1366. doi: 10.1016/j.camwa.2019.03.013.

show all references

References:

[1]	M. Allen, L. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Arch. Ration. Mech. Anal., 221 (2016), 603-630. doi: 10.1007/s00205-016-0969-z.
[2]	M. Chipot, V. Valente and G. Vergara Caffarelli, Remarks on a nonlocal problems involving the Dirichlet energy, Rend. Semin. Mat. Univ. Padova, 110 (2003), 199-220. doi: 10.5167/uzh-21865.
[3]	D. del-Castillo-Negrete, B. A. Carreras and V. E. Lynch, Fractional diffusion in plasma turbulance,, Phys. Plasmas, 11 (2004), 3854-3864. doi: 10.1063/1.1767097.
[4]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[5]	H. Ding and J. Zhou, Local existence, global existence and blow-up of solutions to a nonlocal Kirchhoff diffusion problem, Nonlinearity, 33 (2020), 1046-1063. doi: 10.1088/1361-6544/ab5920.
[6]	Y. Q. Fu and M. Q. Xiang, The existence of weak solutions for parabolic variational inequalities with $(p(x,t),q(x,t))$-growth, Appl. Anal., 93 (2014), 65-83. doi: 10.1080/00036811.2012.755735.
[7]	Y. Q. Fu, On potential wells and vacuum isolating of solutions for space-fractional wave equations,, Adv. Differential Equations and Control Processes, 18 (2017), 149-176. doi: 10.17654/DE018030149.
[8]	R. Gorenflo, Y. Luchko and M. Yamamoto, Time-Fractional diffusion equation in the fractional Sobolev spaces,, Fract. Calc. Appl. Anal., 18 (2015), 799-820. doi: 10.1515/fca-2015-0048.
[9]	B. L. Guo, X. K. Pu and F. H. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, Originally published by Science Press in 2011. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9543.
[10]	P. Hartman, Ordinary Differential Equations, 2$^nd$ edition, Society for Industrial and Applied Mathematics, Philadelphia, 1982. doi: 10.1137/1.9780898719222.
[11]	Y. Z. Han and Q. W. Li, Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Comput. Math. Appl., 75 (2018), 3283-3297. doi: 10.1016/j.camwa.2018.01.047.
[12]	Y. Z. Han, W. J. Gao, Z. Sun and H. X. Li, Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy, Comput. Math. Appl., 76 (2018), 2477-2483. doi: 10.1016/j.camwa.2018.08.043.
[13]	J. X. Jia, J. G. Peng and J. Q. Yang, Harnack's inequality for a space-time fractional diffusion equation and applications to an inverse source problem, J. Differential Equations, 262 (2017), 4415-4450. doi: 10.1016/j.jde.2017.01.002.
[14]	V. N. Kolokoltsov and M. A. Veretennikov, Well-posedness and regularity of the cauchy problem for nonlinear fractional in time and space equations,, Fract. Differ. Calc., 4 (2014), 1-30. doi: 10.7153/fdc-04-01.
[15]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[16]	J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod Gauthier-Villars, Paris, 1969.
[17]	L. Li and J. G. Liu, Some compactness criteria for weak solutions of time fractional PDEs, SIAM J. Math. Anal., 50 (2018), 3963-3995. doi: 10.1137/17M1145549.
[18]	L. Li, J. G. Liu and L. Z. Wang, Cauchy problems for Keller-Seqel type time-space fractional diffusion equation, J. Differential Equations, 265 (2018), 1044-1096. doi: 10.1016/j.jde.2018.03.025.
[19]	E. Nane, Fractional cauchy problems on bounded domains: Survey of recent results, in Fractional Dynamics and Control, Springer, (2012), 185-198. doi: 10.1007/978-1-4614-0457-6_15.
[20]	P. Pucci, M. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional $p$-Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806. doi: 10.1007/s00526-015-0883-5.
[21]	P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22. doi: 10.4171/RMI/879.
[22]	N. Pan, B. L. Zhang and J. Cao, Degenerate Kirchhoff-type diffusion problems involving the fractional $p$-Laplacian, Nonlinear Anal. Real World Appl., 37 (2017), 56-70. doi: 10.1016/j.nonrwa.2017.02.004.
[23]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.
[24]	R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2015-2137. doi: 10.3934/dcds.2013.33.2105.
[25]	V. Vergara and R. Zacher, Lyapunov functions and convergence to steady state for differential equations of fractional order, Math. Z., 259 (2008), 287-309. doi: 10.1007/s00209-007-0225-1.
[26]	V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods,, SIAM J. Math. Anal., 47 (2015), 210-239. doi: 10.1137/130941900.
[27]	M. Q. Xiang and Y. Q. Fu, Weak solutions for nonlocal evolutional inequalities involving gradient constraints and variable exponent, Electron. J. Differential Equations, 2013 (2013), 1-17. doi: 10.1186/1687-2770-2013-96.
[28]	M. Q. Xiang, V. D. Rǎdulescu and B. L. Zhang, Nonlocal Kirchhoff diffusion problems: Local existence and blow-up of solutions, Nonlinearity, 31 (2018), 3228-3250. doi: 10.1088/1361-6544/aaba35.
[29]	M. Q. Xiang, B. L. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional $p$-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041. doi: 10.1016/j.jmaa.2014.11.055.
[30]	R. Zacher, Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients, J. Math. Anal. Appl., 348 (2008), 137-149. doi: 10.1016/j.jmaa.2008.06.054.
[31]	R. Zacher, Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces, Funkc. Ekvac., 52 (2009), 1-18. doi: 10.1619/fesi.52.1.
[32]	R. Zacher, A De Giorgi-Nash type theorem for time fractional diffusion equations,, Math. Ann., 356 (2013), 99-146. doi: 10.1007/s00208-012-0834-9.
[33]	Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations,, Comput. Math. Appl., 59 (2010), 1063-1077. doi: 10.1016/j.camwa.2009.06.026.
[34]	Q. G. Zhang, H. R. Sun and Y. N. Li, Global existence and blow-up of solutions of Cauchy problems for a time fractional diffusion system, Comput. Math. Appl., 78 (2019), 1357-1366. doi: 10.1016/j.camwa.2019.03.013.

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