# American Institute of Mathematical Sciences

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

## Well-posedness of abstract distributed-order fractional diffusion equations

 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.
Citation: Junxiong Jia, Jigen Peng, Kexue Li. Well-posedness of abstract distributed-order fractional diffusion equations. Communications on Pure and 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. [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. [3] E. Bazhlekova, "Fractional Evolution Equations in Banach Spaces," Ph.D thesis, Eindhoven University of Technology, 2001. [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. [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. [6] C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum, 80 (2010), 121-142. doi: 10.1007/s00233-009-9184-7. [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. [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. [9] A. Einstein, "Investigations on the Theory of the Brownian Movement," Dover Publications Inc., New York, 1956. [10] R. Gorenflo and F. Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal., 1 (1998), 167-191. [11] T. Kato, "Perturbation Theory for Linear Operators," Springer, Berlin, 1966. doi: 10.1007/978-3-642-53393-8. [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. [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. [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. [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. [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. [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. [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. [19] Celso. M. Carracedo and Miguel. S. Alix, "The Theory of Fractional Powers of Operators," North-Holland Publishing Co., Amsterdam, 2001. [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. [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. [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. [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. [24] M. Naber, Distributed order fractional subdiffusion, Fractals, 12 (2004), 23-32. doi: 10.1142/S0218348X04002410. [25] I. Podlubny, "Fractional Differential Equations," Academic Press, New York, 1999. [26] J. Pr$\ddotu$ss, "Evolutionary Integral Equations and Applications," Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6. [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. [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. [29] Igor. M. Sokolov, Aleksei. V. Chechkin and J. Klafter, Distributed-order fractional kinetics, Acta Phys. Polon., 35 (2004), 1323-1341. [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.

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##### References:
 [1] D. Applebaum, "Lévy Processes and Stochastic Calculus," Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511755323. [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. [3] E. Bazhlekova, "Fractional Evolution Equations in Banach Spaces," Ph.D thesis, Eindhoven University of Technology, 2001. [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. [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. [6] C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum, 80 (2010), 121-142. doi: 10.1007/s00233-009-9184-7. [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. [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. [9] A. Einstein, "Investigations on the Theory of the Brownian Movement," Dover Publications Inc., New York, 1956. [10] R. Gorenflo and F. Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal., 1 (1998), 167-191. [11] T. Kato, "Perturbation Theory for Linear Operators," Springer, Berlin, 1966. doi: 10.1007/978-3-642-53393-8. [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. [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. [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. [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. [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. [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. [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. [19] Celso. M. Carracedo and Miguel. S. Alix, "The Theory of Fractional Powers of Operators," North-Holland Publishing Co., Amsterdam, 2001. [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. [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. [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. [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. [24] M. Naber, Distributed order fractional subdiffusion, Fractals, 12 (2004), 23-32. doi: 10.1142/S0218348X04002410. [25] I. Podlubny, "Fractional Differential Equations," Academic Press, New York, 1999. [26] J. Pr$\ddotu$ss, "Evolutionary Integral Equations and Applications," Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6. [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. [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. [29] Igor. M. Sokolov, Aleksei. V. Chechkin and J. Klafter, Distributed-order fractional kinetics, Acta Phys. Polon., 35 (2004), 1323-1341. [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.
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