非自治Caputo分数阶发展方程弱解的适定性与不变集

数学物理学报 ›› 2021, Vol. 41 ›› Issue (1): 149-165.

非自治Caputo分数阶发展方程弱解的适定性与不变集

西宣宣,侯咪咪,周先锋*()

安徽大学数学科学学院合肥 230039

收稿日期:2019-11-19 出版日期:2021-02-26 发布日期:2021-01-29
通讯作者: 周先锋 E-mail:zhouxf@ahu.edu.cn
基金资助:
国家自然科学基金(11471015);国家自然科学基金(11371027);国家自然科学基金(11601003);安徽省自然科学基金(1508085MA01);安徽省自然科学基金(1608085MA12);安徽省自然科学基金(1708085MA15);安徽省高校自然科学研究项目(KJ2016A023)

Invariance Sets and Well-Posedness for the Weak Solution of Non-Autonomous Caputo Fractional Evolution Equation

Xuanxuan Xi,Mimi Hou,Xianfeng Zhou*()

School of Mathematical Sciences, Anhui University, Hefei 230039

Received:2019-11-19 Online:2021-02-26 Published:2021-01-29
Contact: Xianfeng Zhou E-mail:zhouxf@ahu.edu.cn
Supported by:
the NSFC(11471015);the NSFC(11371027);the NSFC(11601003);the NSF of Anhui Province(1508085MA01);the NSF of Anhui Province(1608085MA12);the NSF of Anhui Province(1708085MA15);the Program of Natural Science Research for Universities of Anhui Province(KJ2016A023)

摘要/Abstract

摘要：

该文分析一族含有依赖时间参数t线性算子的时间分数阶非自治发展方程，利用Lions表示定理，得到了弱解适定性的充分条件; 基于正交投影，建立了时间分数阶发展方程弱解的不变性准则.该文所研究方程中的算子是依赖时间的.

关键词: 适定性, Caputo分数阶导数, 弱解, 非自治发展方程

Abstract:

The purpose of this paper is to analyze the time-fractional non-autonomous evolution equation which is associated with a family of linear operators depending on the time parameter $t$. Using the representation theorem by Lions, we obtain the sufficient conditions for well-posedness of the weak solution. Based on an orthogonal projection, we establish the invariance criterion for the weak solution of the time-fractional evolution equation. The operators of investigated equations are time-dependent.

Key words: Well-posedness, Caputo fractional derivative, Weak solution, Non-autonomous evolution equation

中图分类号:

O175

西宣宣,侯咪咪,周先锋. 非自治Caputo分数阶发展方程弱解的适定性与不变集[J]. 数学物理学报, 2021, 41(1): 149-165.

Xuanxuan Xi,Mimi Hou,Xianfeng Zhou. Invariance Sets and Well-Posedness for the Weak Solution of Non-Autonomous Caputo Fractional Evolution Equation[J]. Acta mathematica scientia,Series A, 2021, 41(1): 149-165.

参考文献 29

1	Podlubny I . Fractional Differential Equations. New York: Academic Press, 1993
2	Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier Science B V, 2006
3	Zhou Y. Basic Theory of Fractional Differential Equations. Singapore: World Scientific, 2014
4	Elsayed E M , Harikrishnan S , Kanagarajan K . On the existence and stability of boundary value problem for differential equation with Hilfer-Katugampola fractional derivative. Acta Math Sci, 2019, 39: 1568- 1578 doi: 10.1007/s10473-019-0608-5
5	Migorski S , Zeng S D . Mixed variational inequalities driven by fractional evolutionary equations. Acta Math Sci, 2019, 39: 461- 468 doi: 10.1007/s10473-019-0211-9
6	Cao J X , Song G J , Wang J , Shi Q , Sun S . Blow-up and global solutions for a class of time fractional nonlinear reaction-diffusion equation with weakly spatial source. Appl Math Lett, 2019, 91: 201- 206 doi: 10.1016/j.aml.2018.12.020
7	Guo S , Mei L Q , Zhang Z Q , Chen J , He Y , Li Y . Finite difference/Hermite-Galerkin spectral method for multi-dimensional time-fractional nonlinear reaction-diffusion equation in unbounded domains. Appl Math Model, 2019, 70: 246- 263 doi: 10.1016/j.apm.2019.01.018
8	Taki-Eddine O , Abdelfatah B . A priori estimates for weak solution for a time-fractional nonlinear reaction-diffusion equations with an integral condition. Chaos Soliton Fract, 2017, 103: 79- 89 doi: 10.1016/j.chaos.2017.05.035
9	Mu J , Ahmad B , Huang S . Existence and regularity of solutions to time-fractional diffusion equations. Comput Math Appl, 2017, 73: 985- 996 doi: 10.1016/j.camwa.2016.04.039
10	Zhou Y , Shangerganesh L , Manimaran J , Debbouche A . A class of time-fractional reaction-diffusion equation with nonlocal boundary condition. Math Meth Appl Sci, 2018, 41: 2987- 2999 doi: 10.1002/mma.4796
11	Kubica A , Yamamoto M . Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients. Fract Calc Appl Anal, 2018, 21: 276- 311 doi: 10.1515/fca-2018-0018
12	Dehestani H , Ordokhani Y , Razzaghi M . Fractional-order Legendre-Laguerre functions and their applications in fractional partial differential equations. Appl Math Comput, 2018, 336: 433- 453
13	Ayouch C , Essoufi E , Tilioua M . Global weak solutions to a spatio-temporal fractional Landau-Lifshitz-Bloch equation. Comput Math Appl, 2019, 77: 1347- 1357 doi: 10.1016/j.camwa.2018.11.016
14	Lions J L . Equations Différentielles Opérationnelles et Problèmes aux Limits. Heidelberg: Springer, 1961
15	Thomaschewski S. Form methods for autonomous and non-autonomous Cauchy problems[D]. Universität Ulm, 2003
16	Fackler S. J.L.Lions' problem concerning maximal regularity of equations governed by non-autonomous forms. Ann I H Poincare-An, 2017, 34: 699-709
17	Achache M , Ouhabaz E M . Lions' maximal regularity problem with $H^{\frac{1}{2}}$-regularity. J Differential Equations, 2019, 266: 3654- 3678 doi: 10.1016/j.jde.2018.09.015
18	Arendt W , Dier D . Rection-diffusion systems governed by non-autonomous forms. J Differential Equations, 2018, 264: 6362- 6379 doi: 10.1016/j.jde.2018.01.035
19	Ouhabaz E M . Invariance of closed convex sets and domination criteria for semigroups. Potential Anal, 1996, 5: 611- 625 doi: 10.1007/BF00275797
20	Arendt W , Dier D , Ouhabaz E M . Invariance of convex sets for non-autonomous evolution equations governed by forms. J London Math Soc, 2014, 89: 903- 916 doi: 10.1112/jlms/jdt082
21	Dier D . Non-autonomous forms and invariance. Math Nachr, 2019, 292: 603- 614 doi: 10.1002/mana.201700090
22	Zhou Y , Peng L . Weak solutions of the time-fractional Navier-Stokes equations and optimal control. Comput Math Appl, 2017, 73: 1016- 1027 doi: 10.1016/j.camwa.2016.07.007
23	Agrawal O P . Fractional variational calculus in terms of Riesz fractional derivatives. J Phys A, 2007, 40: 6287- 6303 doi: 10.1088/1751-8113/40/24/003
24	Lawrence C E. Partial Differential Equations. Providence, RI: American Mathematical Society, 1997
25	Temam R. Navier Stokes Equations: Theory and Numerical Analysis. Amsterdam: North-Holland, 1977
26	Liu S , Wu X , Zhou X F , Wei J . Asymptotical stability of Riemann-Liouville fractional nonlinear systems. Nonlinear Dyn, 2016, 86: 65- 71 doi: 10.1007/s11071-016-2872-4
27	Ye H , Gao J , Ding Y . A generalized Gronwall inequality and its application to a fractional differential equation. J Math Anal Appl, 2007, 328: 1075- 1081 doi: 10.1016/j.jmaa.2006.05.061
28	Hille E, Phillips R S. Functional Analysis and Semi-groups. Providence, RI: American Mathematical Society, 1957
29	Arendt W , Monniaux S . Maximal regularity for non-autonomous Robin boundary conditions. Math Nachr, 2016, 289: 1325- 1340 doi: 10.1002/mana.201400319

[1]	罗娇,漆前,罗宏. 高阶各向异性Cahn-Hilliard-Navier-Stokes系统的弱解[J]. 数学物理学报, 2020, 40(6): 1599-1611.
[2]	佟玉霞,王薪茹,谷建涛. Orlicz空间中A-调和方程很弱解得L^ϕ估计[J]. 数学物理学报, 2020, 40(6): 1461-1480.
[3]	张雅楠,闫硕,佟玉霞. 自然增长条件下的非齐次A-调和方程弱解的梯度估计[J]. 数学物理学报, 2020, 40(2): 379-394.
[4]	张利民,徐海燕,金春花. 具渗流扩散和间接信号产生的Prey-Taxis模型解的整体存在性与稳定性[J]. 数学物理学报, 2019, 39(6): 1381-1404.
[5]	李凯,杨晗,王凡. 三维带有衰减项的不可压缩磁流体力学方程组弱解与强解的研究[J]. 数学物理学报, 2019, 39(3): 518-528.
[6]	朱波,刘立山. 带瞬时脉冲的分数阶非自制发展方程解的存在唯一性[J]. 数学物理学报, 2019, 39(1): 105-113.
[7]	张申贵. 带类p(x)-拉普拉斯算子的双非局部问题的无穷多解[J]. 数学物理学报, 2018, 38(3): 514-526.
[8]	邢超, 任猛章, 罗宏. 热盐环流方程全局弱解的存在性[J]. 数学物理学报, 2018, 38(2): 284-290.
[9]	蔡钢. Banach空间上有限时滞退化微分方程的适定性[J]. 数学物理学报, 2018, 38(2): 264-275.
[10]	杜广伟, 钮鹏程. 非线性次椭圆方程障碍问题很弱解的高阶可积性[J]. 数学物理学报, 2017, 37(1): 122-145.
[11]	Marko Kostić, 李成刚, 李淼. 抽象多项Riemann-Liouville分数阶微分方程[J]. 数学物理学报, 2016, 36(4): 601-622.
[12]	刘健, 连汝续, 钱茂福. 双极Navier-Stokes-Poisson方程整体解的存在性[J]. 数学物理学报, 2014, 34(4): 960-976.
[13]	曲风龙, 李希亮. 各向同性的麦克斯韦方程的内部传输问题及反射参数的估计[J]. 数学物理学报, 2013, 33(6): 1178-1188.
[14]	朱莉, 夏福全. 广义混合变分不等式的Levitin-Polyak适定性[J]. 数学物理学报, 2012, 32(4): 633-643.
[15]	芮杰, 杨孝平. 极小全变分流第一边值问题弱解的存在性[J]. 数学物理学报, 2011, 31(4): 958-969.