March  2013, 2(1): 173-192. doi: 10.3934/eect.2013.2.173

Approximation of a semigroup model of anomalous diffusion in a bounded set

1. 

Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, United States

2. 

Department of Mathematics and Statistics, University of Maryland Baltimore County (UMBC), Baltimore, MD 21250

Received  November 2012 Revised  December 2012 Published  January 2013

The convergence is established for a sequence of operator semigroups, where the limiting object is the transition semigroup for a reflected stable processes. For semilinear equations involving the generators of these transition semigroups, an approximation method is developed as well. This makes it possible to derive an a priori bound for solutions to these equations, and therefore prove global existence of solutions. An application to epidemiology is also given.
Citation: Stephen Thompson, Thomas I. Seidman. Approximation of a semigroup model of anomalous diffusion in a bounded set. Evolution Equations and Control Theory, 2013, 2 (1) : 173-192. doi: 10.3934/eect.2013.2.173
References:
[1]

K. Bogdan, K. Burdzy and Z. Chen, Censored stable processes, Probab. Theory Relat. Fields, 19 (2003), 89-152. doi: 10.1007/s00440-003-0275-1.

[2]

D. Brockmann, Human mobility and spatial disease dynamics, in "Reviews of Nonlinear Dynamics and Complexity, 2" (ed. H. G. Schuster), Wiley-VCH, (2009), 1-24.

[3]

D. Brockmann, L. Hufnagel and T. Geisel, The scaling laws of human travel, Nature, 439 (2006), 462-465.

[4]

Z. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on d-sets, Stoch. Process. Appl., 108 (2003), 27-62. doi: 10.1016/S0304-4149(03)00105-4.

[5]

Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.

[6]

K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer, New York, 1995.

[7]

M. Fukushima, T. Oshima and M. Takeda, "Dirichlet Forms and Symmetric Markov Processes," Walter de Gruyter, Berlin, 1994. doi: 10.1515/9783110889741.

[8]

P. Grisvard, Caractérisation de quelques espaces d'interpolation, Arch. Rational Mech. Anal., 26 (1967), 431-458.

[9]

Q. Guan and Z. Ma, Reflected symmetric $\alpha$-stable processes and regional fractional laplacian, Probab. Theory Relat. Fields, 134 (2006), 649-694. doi: 10.1007/s00440-005-0438-3.

[10]

K. Gustafson and G. Lumer, Multiplicative perturbation of semigroup generators, Pac. J. Math., 41 (1972), 731-742.

[11]

E. Hanert, Front dynamics in a two-species competition model driven by Lévy flights, J. Theor. Biol., 300 (2012), 134-142. doi: 10.1016/j.jtbi.2012.01.022.

[12]

E. Hanert, E. Schumacher and E. Eleersnijder, Front dynamics in fractional-order epidemic models, J. Theor. Biol., 279 (2011), 9-16.

[13]

K. Ito and F. Kappel, The trotter kato theorem and approximation of PDEs, Math. Comput., 67 (1998), 21-44. doi: 10.1090/S0025-5718-98-00915-6.

[14]

P. Kim, Weak convergence of censored and reflected stable processes, Stoch. Process. Appl., 116 (2006), 1792-1814. doi: 10.1016/j.spa.2006.04.006.

[15]

R. Klages, G. Radons and I. M. Sokolov, "Anomalous Transport," Wiley-VCH, Weinheim 2008.

[16]

L. Lorenzi, A. Lundardi, G. Metafune and D. Pallara, "Analytic Semigroups and Reaction-Diffusion Problems," unpublished Lecture Notes, http://www.math.unipr.it/~lunardi/LectureNotes/I-Sem2005.pdf.

[17]

T. Lux and M. Marchesi, Scaling and criticality in a stochastic multi-agent model of a financial market, Nature, 397 (1999), 498-500.

[18]

G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. G. E. da Luz, E. P. Raposo and H. E. Stanley, Optimizing the success of random searches, Nature, 401 (1999), 911-914.

[19]

J. Wloka, "Partial Differential Equations," Cambridge University Press, London 1987.

[20]

M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries: Analysis, Differential Calculus, and Optimization," Society for Industrial and Applied Mathematics, Philadelphia 2001.

show all references

References:
[1]

K. Bogdan, K. Burdzy and Z. Chen, Censored stable processes, Probab. Theory Relat. Fields, 19 (2003), 89-152. doi: 10.1007/s00440-003-0275-1.

[2]

D. Brockmann, Human mobility and spatial disease dynamics, in "Reviews of Nonlinear Dynamics and Complexity, 2" (ed. H. G. Schuster), Wiley-VCH, (2009), 1-24.

[3]

D. Brockmann, L. Hufnagel and T. Geisel, The scaling laws of human travel, Nature, 439 (2006), 462-465.

[4]

Z. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on d-sets, Stoch. Process. Appl., 108 (2003), 27-62. doi: 10.1016/S0304-4149(03)00105-4.

[5]

Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.

[6]

K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer, New York, 1995.

[7]

M. Fukushima, T. Oshima and M. Takeda, "Dirichlet Forms and Symmetric Markov Processes," Walter de Gruyter, Berlin, 1994. doi: 10.1515/9783110889741.

[8]

P. Grisvard, Caractérisation de quelques espaces d'interpolation, Arch. Rational Mech. Anal., 26 (1967), 431-458.

[9]

Q. Guan and Z. Ma, Reflected symmetric $\alpha$-stable processes and regional fractional laplacian, Probab. Theory Relat. Fields, 134 (2006), 649-694. doi: 10.1007/s00440-005-0438-3.

[10]

K. Gustafson and G. Lumer, Multiplicative perturbation of semigroup generators, Pac. J. Math., 41 (1972), 731-742.

[11]

E. Hanert, Front dynamics in a two-species competition model driven by Lévy flights, J. Theor. Biol., 300 (2012), 134-142. doi: 10.1016/j.jtbi.2012.01.022.

[12]

E. Hanert, E. Schumacher and E. Eleersnijder, Front dynamics in fractional-order epidemic models, J. Theor. Biol., 279 (2011), 9-16.

[13]

K. Ito and F. Kappel, The trotter kato theorem and approximation of PDEs, Math. Comput., 67 (1998), 21-44. doi: 10.1090/S0025-5718-98-00915-6.

[14]

P. Kim, Weak convergence of censored and reflected stable processes, Stoch. Process. Appl., 116 (2006), 1792-1814. doi: 10.1016/j.spa.2006.04.006.

[15]

R. Klages, G. Radons and I. M. Sokolov, "Anomalous Transport," Wiley-VCH, Weinheim 2008.

[16]

L. Lorenzi, A. Lundardi, G. Metafune and D. Pallara, "Analytic Semigroups and Reaction-Diffusion Problems," unpublished Lecture Notes, http://www.math.unipr.it/~lunardi/LectureNotes/I-Sem2005.pdf.

[17]

T. Lux and M. Marchesi, Scaling and criticality in a stochastic multi-agent model of a financial market, Nature, 397 (1999), 498-500.

[18]

G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. G. E. da Luz, E. P. Raposo and H. E. Stanley, Optimizing the success of random searches, Nature, 401 (1999), 911-914.

[19]

J. Wloka, "Partial Differential Equations," Cambridge University Press, London 1987.

[20]

M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries: Analysis, Differential Calculus, and Optimization," Society for Industrial and Applied Mathematics, Philadelphia 2001.

[1]

Pu-Zhao Kow, Masato Kimura. The Lewy-Stampacchia inequality for the fractional Laplacian and its application to anomalous unidirectional diffusion equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 2935-2957. doi: 10.3934/dcdsb.2021167

[2]

Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236

[3]

Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121

[4]

Yixuan Wu, Yanzhi Zhang. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 851-876. doi: 10.3934/dcdss.2022016

[5]

Valentin Keyantuo, Louis Tebou, Mahamadi Warma. A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2875-2889. doi: 10.3934/dcds.2020152

[6]

Isabelle Kuhwald, Ilya Pavlyukevich. Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3175-3190. doi: 10.3934/dcdsb.2016092

[7]

Christian Licht, Thibaut Weller. Approximation of semi-groups in the sense of Trotter and asymptotic mathematical modeling in physics of continuous media. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1709-1741. doi: 10.3934/dcdss.2019114

[8]

Hayat Zouiten, Ali Boutoulout, Delfim F. M. Torres. Regional enlarged observability of Caputo fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 1017-1029. doi: 10.3934/dcdss.2020060

[9]

Yulong Li, Aleksey S. Telyakovskiy, Emine Çelik. Analysis of one-sided 1-D fractional diffusion operator. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1673-1690. doi: 10.3934/cpaa.2022039

[10]

Yuanwei Qi. Anomalous exponents and RG for nonlinear diffusion equations. Conference Publications, 2005, 2005 (Special) : 738-745. doi: 10.3934/proc.2005.2005.738

[11]

Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248

[12]

Brandy Rapatski, Petra Klepac, Stephen Dueck, Maoxing Liu, Leda Ivic Weiss. Mathematical epidemiology of HIV/AIDS in cuba during the period 1986-2000. Mathematical Biosciences & Engineering, 2006, 3 (3) : 545-556. doi: 10.3934/mbe.2006.3.545

[13]

Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control and Related Fields, 2022, 12 (1) : 115-146. doi: 10.3934/mcrf.2021004

[14]

Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171

[15]

Juan-Luis Vázquez. Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 857-885. doi: 10.3934/dcdss.2014.7.857

[16]

Siwei Duo, Hong Wang, Yanzhi Zhang. A comparative study on nonlocal diffusion operators related to the fractional Laplacian. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 231-256. doi: 10.3934/dcdsb.2018110

[17]

Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003

[18]

Hongjie Li. Recent progress on the mathematical study of anomalous localized resonance in elasticity. Electronic Research Archive, 2020, 28 (3) : 1257-1272. doi: 10.3934/era.2020069

[19]

Ravi Shanker Dubey, Pranay Goswami. Mathematical model of diabetes and its complication involving fractional operator without singular kernal. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2151-2161. doi: 10.3934/dcdss.2020144

[20]

Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3387-3399. doi: 10.3934/dcdss.2021017

2021 Impact Factor: 1.169

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]