# American Institue of Mathematical Sciences

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 & Control Theory, 2013, 2 (1) : 173-192. doi: 10.3934/eect.2013.2.173
##### References:
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##### References:
 [1] K. Bogdan, K. Burdzy and Z. Chen, Censored stable processes,, Probab. Theory Relat. Fields, 19 (2003), 89. doi: 10.1007/s00440-003-0275-1. [2] D. Brockmann, Human mobility and spatial disease dynamics,, in, 2 (2009), 1. [3] D. Brockmann, L. Hufnagel and T. Geisel, The scaling laws of human travel,, Nature, 439 (2006), 462. [4] Z. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on d-sets,, Stoch. Process. Appl., 108 (2003), 27. 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. [6] K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Springer, (1995). [7] M. Fukushima, T. Oshima and M. Takeda, "Dirichlet Forms and Symmetric Markov Processes,", Walter de Gruyter, (1994). doi: 10.1515/9783110889741. [8] P. Grisvard, Caractérisation de quelques espaces d'interpolation,, Arch. Rational Mech. Anal., 26 (1967), 431. [9] Q. Guan and Z. Ma, Reflected symmetric $\alpha$-stable processes and regional fractional laplacian,, Probab. Theory Relat. Fields, 134 (2006), 649. doi: 10.1007/s00440-005-0438-3. [10] K. Gustafson and G. Lumer, Multiplicative perturbation of semigroup generators,, Pac. J. Math., 41 (1972), 731. [11] E. Hanert, Front dynamics in a two-species competition model driven by Lévy flights,, J. Theor. Biol., 300 (2012), 134. 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. [13] K. Ito and F. Kappel, The trotter kato theorem and approximation of PDEs,, Math. Comput., 67 (1998), 21. 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. doi: 10.1016/j.spa.2006.04.006. [15] R. Klages, G. Radons and I. M. Sokolov, "Anomalous Transport,", Wiley-VCH, (2008). [16] L. Lorenzi, A. Lundardi, G. Metafune and D. Pallara, "Analytic Semigroups and Reaction-Diffusion Problems,", unpublished Lecture Notes, (). [17] T. Lux and M. Marchesi, Scaling and criticality in a stochastic multi-agent model of a financial market,, Nature, 397 (1999), 498. [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. [19] J. Wloka, "Partial Differential Equations,", Cambridge University Press, (1987). [20] M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries: Analysis, Differential Calculus, and Optimization,", Society for Industrial and Applied Mathematics, (2001).
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