American Institute of Mathematical Sciences

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March  2014, 13(2): 605-621. doi: 10.3934/cpaa.2014.13.605

Well-posedness of abstract distributed-order fractional diffusion equations

Junxiong Jia 1, , Jigen Peng 1, and  Kexue Li 1,

1. 

School of Mathematics and Statistics, Ministry of Education Key Lab for Intelligent Networks and Network Security, Xi'an Jiaotong University, Xi'an 710049, China, China, China

Received  December 2012 Revised  August 2013 Published  October 2013

In this paper, based on distributed-order fractional diffusion equation we propose the distributed-order fractional abstract Cauchy problem (DFACP) and study the well-posedness of DFACP. Using functional calculus technique, we prove that the general distributed-order fractional operator generates a bounded analytic $\alpha$-times resolvent operator family or a $C_0$-semigroup under some suitable conditions. In addition, we reveal the relation between two $\alpha$-times resolvent families generated by the sectorial operator $A$ and the special distributed-order fractional operator, $p_1A^{\beta_1}+ p_2A^{\beta_2}+\ldots +p_nA^{\beta_n}$, respectively.
Keywords: Distributed-order fractional equation, fractional power., $\alpha$-times resolvent family, sectorial operator.
Mathematics Subject Classification: Primary: 26A33; Secondary: 47D0.
Citation: Junxiong Jia, Jigen Peng, Kexue Li. Well-posedness of abstract distributed-order fractional diffusion equations. Communications on Pure & Applied Analysis, 2014, 13 (2) : 605-621. doi: 10.3934/cpaa.2014.13.605
References:
[1]

D. Applebaum, "Lévy Processes and Stochastic Calculus," Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511755323.  Google Scholar

[2]

R. Balescu, Anomalous transport in turbulent plasmas and continuous time random walks, Phys. Rev. E, 51 (1995), 4807-4822. doi: 10.1103/PhysRevE.51.4807.  Google Scholar

[3]

E. Bazhlekova, "Fractional Evolution Equations in Banach Spaces," Ph.D thesis, Eindhoven University of Technology, 2001.  Google Scholar

[4]

Alfred. S. Carasso and T. Kato, On subordinated holomorphic semigroups, Trans. Amer. Math. Soc., 327 (1991), 867-878. doi: 10.1090/S0002-9947-1991-1018572-4.  Google Scholar

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Aleksei. V. Chechkin, R. Gorenflo and Igor. M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E, 66 (2002), 046129. doi: 10.1103/PhysRevE.66.046129.  Google Scholar

[6]

C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum, 80 (2010), 121-142. doi: 10.1007/s00233-009-9184-7.  Google Scholar

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G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201. doi: 10.1007/BF01163654.  Google Scholar

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N. Dungey, Asymptotic type for sectorial operators and an integral of fractional powers, J. Func. Anal., 256 (2009), 1387-1407. doi: 10.1016/j.jfa.2008.07.020.  Google Scholar

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A. Einstein, "Investigations on the Theory of the Brownian Movement," Dover Publications Inc., New York, 1956.  Google Scholar

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R. Gorenflo and F. Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal., 1 (1998), 167-191.  Google Scholar

[11]

T. Kato, "Perturbation Theory for Linear Operators," Springer, Berlin, 1966. doi: 10.1007/978-3-642-53393-8.  Google Scholar

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A. Klemm, Hans. P. Müller and R. Kimmich, NMR microscopy of pore-space backbones in rock, sponge, and sand in comparison with random percolation model objects, Phys. Rev. E, 55 (1997), 4413-4422. doi: 10.1103/PhysRevE.55.4413.  Google Scholar

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M. Haase, "The Functional Calculus for Sectorial Operators," Operator Theory: Advances and Applications, Birkh$\ddota$uzer, 2006. doi: 10.1007/3-7643-7698-8.  Google Scholar

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M. Li, C. Chen and F. Li, On fractional powers of generators of fractional resolvent families, J. Func. Anal., 259 (2010), 2702-2726. doi: 10.1016/j.jfa.2010.07.007.  Google Scholar

[15]

K. Li and J. Jia, Existence and uniqueness of mild solutions for abstract delay fractional differential equations, Comput. Math. Appl., 63 (2011), 1398-1404. doi: 10.1016/j.camwa.2011.02.038.  Google Scholar

[16]

F. Mainardi, G. Pagnini and R. Gorenflo, Some aspects of fractional diffusion equations of single and distributed order, Appl. Math. Comput., 187 (2007), 295-305. doi: 10.1016/j.amc.2006.08.126.  Google Scholar

[17]

F. Mainardi, Y. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4 (2001), 153-192.  Google Scholar

[18]

Rosario. N. Mantegna and Harry. E. Stanley, Stochastic process with ultraslow convergence to a gaussian: the truncated Levy flight, Phys. Rev. Lett., 73 (1994), 2946-2949. doi: 10.1103/PhysRevLett.73.2946.  Google Scholar

[19]

Celso. M. Carracedo and Miguel. S. Alix, "The Theory of Fractional Powers of Operators," North-Holland Publishing Co., Amsterdam, 2001.  Google Scholar

[20]

Mark. M. Meerschaert and Hans. P. Scheffler, Stochastic model for ultraslow diffusion, Stoch. Proc. Appl., 116 (2006), 1215-1235. doi: 10.1016/j.spa.2006.01.006.  Google Scholar

[21]

Mark. M. Meerschaert, E. Nane and P. Vellaisamy, Distributed-order fractional diffusions on bounded domains, J. Math. Anal. Appl., 379 (2011), 216-228. doi: 10.1016/j.jmaa.2010.12.056.  Google Scholar

[22]

R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[23]

Elliott. W. Montroll and George. H. Weiss, Random walks on lattices. II, J. Math. Phys., 167 (1965), 167-181. doi: 10.1063/1.1704269.  Google Scholar

[24]

M. Naber, Distributed order fractional subdiffusion, Fractals, 12 (2004), 23-32. doi: 10.1142/S0218348X04002410.  Google Scholar

[25]

I. Podlubny, "Fractional Differential Equations," Academic Press, New York, 1999.  Google Scholar

[26]

J. Pr$\ddotu$ss, "Evolutionary Integral Equations and Applications," Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[27]

Lewis. F. Richardson, Atmospheric diffusion shown on a distance-neighbout graph, Proc. Roy. Soc., 110 (1926), 709-737. doi: 10.1098/rspa.1926.0043.  Google Scholar

[28]

Yakov. G. Sinai, The limiting behaviour of a one-dimensional random walk in a random medium, Theor. Prob. Appl., 27 (1982), 256-268. doi: 10.1137/1127028.  Google Scholar

[29]

Igor. M. Sokolov, Aleksei. V. Chechkin and J. Klafter, Distributed-order fractional kinetics, Acta Phys. Polon., 35 (2004), 1323-1341. Google Scholar

[30]

G. Zumofen and J. Klafter, Spectral random walk of a single molecule, Chem. Phys. Lett., 219 (1994), 303-309. doi: 10.1016/0009-2614(94)87062-4.  Google Scholar

show all references

References:
[1]

D. Applebaum, "Lévy Processes and Stochastic Calculus," Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511755323.  Google Scholar

[2]

R. Balescu, Anomalous transport in turbulent plasmas and continuous time random walks, Phys. Rev. E, 51 (1995), 4807-4822. doi: 10.1103/PhysRevE.51.4807.  Google Scholar

[3]

E. Bazhlekova, "Fractional Evolution Equations in Banach Spaces," Ph.D thesis, Eindhoven University of Technology, 2001.  Google Scholar

[4]

Alfred. S. Carasso and T. Kato, On subordinated holomorphic semigroups, Trans. Amer. Math. Soc., 327 (1991), 867-878. doi: 10.1090/S0002-9947-1991-1018572-4.  Google Scholar

[5]

Aleksei. V. Chechkin, R. Gorenflo and Igor. M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E, 66 (2002), 046129. doi: 10.1103/PhysRevE.66.046129.  Google Scholar

[6]

C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum, 80 (2010), 121-142. doi: 10.1007/s00233-009-9184-7.  Google Scholar

[7]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201. doi: 10.1007/BF01163654.  Google Scholar

[8]

N. Dungey, Asymptotic type for sectorial operators and an integral of fractional powers, J. Func. Anal., 256 (2009), 1387-1407. doi: 10.1016/j.jfa.2008.07.020.  Google Scholar

[9]

A. Einstein, "Investigations on the Theory of the Brownian Movement," Dover Publications Inc., New York, 1956.  Google Scholar

[10]

R. Gorenflo and F. Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal., 1 (1998), 167-191.  Google Scholar

[11]

T. Kato, "Perturbation Theory for Linear Operators," Springer, Berlin, 1966. doi: 10.1007/978-3-642-53393-8.  Google Scholar

[12]

A. Klemm, Hans. P. Müller and R. Kimmich, NMR microscopy of pore-space backbones in rock, sponge, and sand in comparison with random percolation model objects, Phys. Rev. E, 55 (1997), 4413-4422. doi: 10.1103/PhysRevE.55.4413.  Google Scholar

[13]

M. Haase, "The Functional Calculus for Sectorial Operators," Operator Theory: Advances and Applications, Birkh$\ddota$uzer, 2006. doi: 10.1007/3-7643-7698-8.  Google Scholar

[14]

M. Li, C. Chen and F. Li, On fractional powers of generators of fractional resolvent families, J. Func. Anal., 259 (2010), 2702-2726. doi: 10.1016/j.jfa.2010.07.007.  Google Scholar

[15]

K. Li and J. Jia, Existence and uniqueness of mild solutions for abstract delay fractional differential equations, Comput. Math. Appl., 63 (2011), 1398-1404. doi: 10.1016/j.camwa.2011.02.038.  Google Scholar

[16]

F. Mainardi, G. Pagnini and R. Gorenflo, Some aspects of fractional diffusion equations of single and distributed order, Appl. Math. Comput., 187 (2007), 295-305. doi: 10.1016/j.amc.2006.08.126.  Google Scholar

[17]

F. Mainardi, Y. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4 (2001), 153-192.  Google Scholar

[18]

Rosario. N. Mantegna and Harry. E. Stanley, Stochastic process with ultraslow convergence to a gaussian: the truncated Levy flight, Phys. Rev. Lett., 73 (1994), 2946-2949. doi: 10.1103/PhysRevLett.73.2946.  Google Scholar

[19]

Celso. M. Carracedo and Miguel. S. Alix, "The Theory of Fractional Powers of Operators," North-Holland Publishing Co., Amsterdam, 2001.  Google Scholar

[20]

Mark. M. Meerschaert and Hans. P. Scheffler, Stochastic model for ultraslow diffusion, Stoch. Proc. Appl., 116 (2006), 1215-1235. doi: 10.1016/j.spa.2006.01.006.  Google Scholar

[21]

Mark. M. Meerschaert, E. Nane and P. Vellaisamy, Distributed-order fractional diffusions on bounded domains, J. Math. Anal. Appl., 379 (2011), 216-228. doi: 10.1016/j.jmaa.2010.12.056.  Google Scholar

[22]

R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[23]

Elliott. W. Montroll and George. H. Weiss, Random walks on lattices. II, J. Math. Phys., 167 (1965), 167-181. doi: 10.1063/1.1704269.  Google Scholar

[24]

M. Naber, Distributed order fractional subdiffusion, Fractals, 12 (2004), 23-32. doi: 10.1142/S0218348X04002410.  Google Scholar

[25]

I. Podlubny, "Fractional Differential Equations," Academic Press, New York, 1999.  Google Scholar

[26]

J. Pr$\ddotu$ss, "Evolutionary Integral Equations and Applications," Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[27]

Lewis. F. Richardson, Atmospheric diffusion shown on a distance-neighbout graph, Proc. Roy. Soc., 110 (1926), 709-737. doi: 10.1098/rspa.1926.0043.  Google Scholar

[28]

Yakov. G. Sinai, The limiting behaviour of a one-dimensional random walk in a random medium, Theor. Prob. Appl., 27 (1982), 256-268. doi: 10.1137/1127028.  Google Scholar

[29]

Igor. M. Sokolov, Aleksei. V. Chechkin and J. Klafter, Distributed-order fractional kinetics, Acta Phys. Polon., 35 (2004), 1323-1341. Google Scholar

[30]

G. Zumofen and J. Klafter, Spectral random walk of a single molecule, Chem. Phys. Lett., 219 (1994), 303-309. doi: 10.1016/0009-2614(94)87062-4.  Google Scholar

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