# American Institute of Mathematical Sciences

March  2016, 36(3): 1279-1319. doi: 10.3934/dcds.2016.36.1279

## Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions

 1 Department of Mathematics, Florida International University, Miami, FL, 33199 2 University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377

Received  November 2014 Revised  March 2015 Published  August 2015

We investigate the long term behavior in terms of finite dimensional global attractors and (global) asymptotic stabilization to steady states, as time goes to infinity, of solutions to a non-local semilinear reaction-diffusion equation associated with the fractional Laplace operator on non-smooth domains subject to Dirichlet, fractional Neumann and Robin boundary conditions.
Citation: Ciprian G. Gal, Mahamadi Warma. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1279-1319. doi: 10.3934/dcds.2016.36.1279
##### References:
 [1] S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion,, Physica, A 356 (2005), 403. Google Scholar [2] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory,, Grundlehren der Mathematischen Wissenschaften, (1996). doi: 10.1007/978-3-662-03282-4. Google Scholar [3] N. D. Alikakos, $L^p$-bounds of solutions to reaction-diffusion equations,, Comm. Partial Differential Equations, 4 (1979), 827. doi: 10.1080/03605307908820113. Google Scholar [4] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems,, Mathematical Surveys and Monographs, (2010). doi: 10.1090/surv/165. Google Scholar [5] R. M. Blumenthal and R. K. Getoor, The asymptotic distribution of the eigenvalues for a class of Markov operators,, Pacific J. Math., 9 (1959), 399. doi: 10.2140/pjm.1959.9.399. Google Scholar [6] K. Bogdan, K. Burdzy and Z. Q. Chen, Censored stable processes,, Probab. Theory Related Fields, 127 (2003), 89. doi: 10.1007/s00440-003-0275-1. Google Scholar [7] J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s\uparrow 1$ and applications,, J. Anal. Math., 87 (2002), 77. doi: 10.1007/BF02868470. Google Scholar [8] L. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian,, J. Eur. Math. Soc., 12 (2010), 1151. doi: 10.4171/JEMS/226. Google Scholar [9] L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian,, Invent. Math., 171 (2008), 425. doi: 10.1007/s00222-007-0086-6. Google Scholar [10] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar [11] Z.-Q. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on $d$-sets,, Stochastic Process. Appl., 108 (2003), 27. doi: 10.1016/S0304-4149(03)00105-4. Google Scholar [12] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Amer. Math. Soc., (2002). Google Scholar [13] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems,, Cambridge University Press, (2000). doi: 10.1017/CBO9780511526404. Google Scholar [14] D. Danielli, N. Garofalo and D.-M. Nhieu, Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carath éodory spaces,, Mem. Amer. Math. Soc., 182 (2006). doi: 10.1090/memo/0857. Google Scholar [15] E. B. Davies, Heat Kernels and Spectral Theory,, Cambridge University Press, (1989). doi: 10.1017/CBO9780511566158. Google Scholar [16] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar [17] S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions,, , (). Google Scholar [18] Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws,, Math. Models Methods Appl. Sci., 23 (2013), 493. doi: 10.1142/S0218202512500546. Google Scholar [19] E. Feireisl, F. Issard-Roch and H. Petzeltová, A non-smooth version of the Lojasiewicz-Simon theorem with applications to non-local phase-field systems,, J. Differential Equations, 199 (2004), 1. doi: 10.1016/j.jde.2003.10.026. Google Scholar [20] H. Gajewski and J. Griepentrog, A descent method for the free energy of multicomponent systems,, Discrete Contin. Dyn. Syst., 15 (2006), 505. doi: 10.3934/dcds.2006.15.505. Google Scholar [21] C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions,, J. Differential Equations, 253 (2012), 126. doi: 10.1016/j.jde.2012.02.010. Google Scholar [22] C. G. Gal, Sharp estimates for the global attractor of scalar reaction-diffusion equations with a Wentzell boundary condition,, J. Nonlinear Science, 22 (2012), 85. doi: 10.1007/s00332-011-9109-y. Google Scholar [23] C. G. Gal and M. Grasselli, Longtime behavior of nonlocal Cahn-Hilliard equations,, Discrete Contin. Dyn. Syst., 34 (2014), 145. doi: 10.3934/dcds.2014.34.145. Google Scholar [24] C. G. Gal and M. Warma, Long-term behavior of reaction-diffusion equations with nonlocal boundary conditions on rough domains,, submitted., (). Google Scholar [25] C. G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractionalp-Laplacian,, submitted., (). Google Scholar [26] A. Garroni and S. Müller, A variational model for dislocations in the line tension limit,, Arch. Ration. Mech. Anal., 181 (2006), 535. doi: 10.1007/s00205-006-0432-7. Google Scholar [27] P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985). Google Scholar [28] Q. Y. Guan, Integration by parts formula for regional fractional Laplacian,, Comm. Math. Phys., 266 (2006), 289. doi: 10.1007/s00220-006-0054-9. Google Scholar [29] Q. Y. Guan and Z. M. Ma, Reflected symmetric $\alpha$-stable processes and regional fractional Laplacian,, Probab. Theory Related Fields, 134 (2006), 649. doi: 10.1007/s00440-005-0438-3. Google Scholar [30] Q. Y. Guan and Z. M. Ma, Boundary problems for fractional Laplacians,, Stoch. Dyn., 5 (2005), 385. doi: 10.1142/S021949370500150X. Google Scholar [31] C. Gui and M. Zhao, Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian,, Annales de l'Institut Henri Poincaré Non Linear Analysis, (2014). doi: 10.1016/j.anihpc.2014.03.005. Google Scholar [32] M. Gunzburger and R. B. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems,, Multiscale Model. Simul., 8 (2010), 1581. doi: 10.1137/090766607. Google Scholar [33] M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps,, Comm. Pure Appl. Math., 62 (2009), 198. doi: 10.1002/cpa.20253. Google Scholar [34] A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbb R^N$,, Math. Rep., 2 (1984). Google Scholar [35] S. Kaplan, On the growth of solutions of quasilinear parabolic equations,, Comm. Pure Appl. Math., 16 (1963), 305. doi: 10.1002/cpa.3160160307. Google Scholar [36] M. Koslowski, A. M. Cuitino and M. Ortiz, A phasefield theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystal,, J. Mech. Phys. Solids, 50 (2002), 2597. doi: 10.1016/S0022-5096(02)00037-6. Google Scholar [37] H. A. Levine and L. E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time,, J. Differential Equations, 16 (1974), 319. doi: 10.1016/0022-0396(74)90018-7. Google Scholar [38] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I,, Springer-Verlag, (1972). Google Scholar [39] S.-O. Londen and H. Petzeltová, Convergence of solutions of a non-local phase-field system,, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 653. doi: 10.3934/dcdss.2011.4.653. Google Scholar [40] A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations,, Arch. Ration. Mech. Anal., 199 (2011), 493. doi: 10.1007/s00205-010-0354-2. Google Scholar [41] Y. Nec, A. A. Nepomnyashchy and A. A. Golovin, Front-type solutions of fractional Allen-Cahn equation,, Phys. D, 237 (2008), 3237. doi: 10.1016/j.physd.2008.08.002. Google Scholar [42] D. Schertzer, M. Larcheveque, J. Duan, V. V. Yanovsky and S. Lovejoy, Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises,, J. Math. Phys., 42 (2001), 200. doi: 10.1063/1.1318734. Google Scholar [43] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,, Amer. Math. Soc., (1997). Google Scholar [44] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary,, J. Math. Pures Appl., 101 (2014), 275. doi: 10.1016/j.matpur.2013.06.003. Google Scholar [45] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, Discrete Contin. Dyn. Syst., 33 (2013), 2105. Google Scholar [46] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar [47] L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous diffusion: A tutorial,, in Order and Chaos (ed. T. Bountis), (2008). Google Scholar [48] M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets,, Potential Anal., 42 (2015), 499. doi: 10.1007/s11118-014-9443-4. Google Scholar [49] M. Warma, Integration by parts formula and regularity of weak solutions of non-local equations involving the fractional $p$-Laplacian with Neumann and Robin boundary conditions on open sets,, preprint., (). Google Scholar

show all references

##### References:
 [1] S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion,, Physica, A 356 (2005), 403. Google Scholar [2] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory,, Grundlehren der Mathematischen Wissenschaften, (1996). doi: 10.1007/978-3-662-03282-4. Google Scholar [3] N. D. Alikakos, $L^p$-bounds of solutions to reaction-diffusion equations,, Comm. Partial Differential Equations, 4 (1979), 827. doi: 10.1080/03605307908820113. Google Scholar [4] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems,, Mathematical Surveys and Monographs, (2010). doi: 10.1090/surv/165. Google Scholar [5] R. M. Blumenthal and R. K. Getoor, The asymptotic distribution of the eigenvalues for a class of Markov operators,, Pacific J. Math., 9 (1959), 399. doi: 10.2140/pjm.1959.9.399. Google Scholar [6] K. Bogdan, K. Burdzy and Z. Q. Chen, Censored stable processes,, Probab. Theory Related Fields, 127 (2003), 89. doi: 10.1007/s00440-003-0275-1. Google Scholar [7] J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s\uparrow 1$ and applications,, J. Anal. Math., 87 (2002), 77. doi: 10.1007/BF02868470. Google Scholar [8] L. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian,, J. Eur. Math. Soc., 12 (2010), 1151. doi: 10.4171/JEMS/226. Google Scholar [9] L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian,, Invent. Math., 171 (2008), 425. doi: 10.1007/s00222-007-0086-6. Google Scholar [10] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar [11] Z.-Q. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on $d$-sets,, Stochastic Process. Appl., 108 (2003), 27. doi: 10.1016/S0304-4149(03)00105-4. Google Scholar [12] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Amer. Math. Soc., (2002). Google Scholar [13] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems,, Cambridge University Press, (2000). doi: 10.1017/CBO9780511526404. Google Scholar [14] D. Danielli, N. Garofalo and D.-M. Nhieu, Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carath éodory spaces,, Mem. Amer. Math. Soc., 182 (2006). doi: 10.1090/memo/0857. Google Scholar [15] E. B. Davies, Heat Kernels and Spectral Theory,, Cambridge University Press, (1989). doi: 10.1017/CBO9780511566158. Google Scholar [16] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar [17] S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions,, , (). Google Scholar [18] Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws,, Math. Models Methods Appl. Sci., 23 (2013), 493. doi: 10.1142/S0218202512500546. Google Scholar [19] E. Feireisl, F. Issard-Roch and H. Petzeltová, A non-smooth version of the Lojasiewicz-Simon theorem with applications to non-local phase-field systems,, J. Differential Equations, 199 (2004), 1. doi: 10.1016/j.jde.2003.10.026. Google Scholar [20] H. Gajewski and J. Griepentrog, A descent method for the free energy of multicomponent systems,, Discrete Contin. Dyn. Syst., 15 (2006), 505. doi: 10.3934/dcds.2006.15.505. Google Scholar [21] C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions,, J. Differential Equations, 253 (2012), 126. doi: 10.1016/j.jde.2012.02.010. Google Scholar [22] C. G. Gal, Sharp estimates for the global attractor of scalar reaction-diffusion equations with a Wentzell boundary condition,, J. Nonlinear Science, 22 (2012), 85. doi: 10.1007/s00332-011-9109-y. Google Scholar [23] C. G. Gal and M. Grasselli, Longtime behavior of nonlocal Cahn-Hilliard equations,, Discrete Contin. Dyn. Syst., 34 (2014), 145. doi: 10.3934/dcds.2014.34.145. Google Scholar [24] C. G. Gal and M. Warma, Long-term behavior of reaction-diffusion equations with nonlocal boundary conditions on rough domains,, submitted., (). Google Scholar [25] C. G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractionalp-Laplacian,, submitted., (). Google Scholar [26] A. Garroni and S. Müller, A variational model for dislocations in the line tension limit,, Arch. Ration. Mech. Anal., 181 (2006), 535. doi: 10.1007/s00205-006-0432-7. Google Scholar [27] P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985). Google Scholar [28] Q. Y. Guan, Integration by parts formula for regional fractional Laplacian,, Comm. Math. Phys., 266 (2006), 289. doi: 10.1007/s00220-006-0054-9. Google Scholar [29] Q. Y. Guan and Z. M. Ma, Reflected symmetric $\alpha$-stable processes and regional fractional Laplacian,, Probab. Theory Related Fields, 134 (2006), 649. doi: 10.1007/s00440-005-0438-3. Google Scholar [30] Q. Y. Guan and Z. M. Ma, Boundary problems for fractional Laplacians,, Stoch. Dyn., 5 (2005), 385. doi: 10.1142/S021949370500150X. Google Scholar [31] C. Gui and M. Zhao, Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian,, Annales de l'Institut Henri Poincaré Non Linear Analysis, (2014). doi: 10.1016/j.anihpc.2014.03.005. Google Scholar [32] M. Gunzburger and R. B. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems,, Multiscale Model. Simul., 8 (2010), 1581. doi: 10.1137/090766607. Google Scholar [33] M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps,, Comm. Pure Appl. Math., 62 (2009), 198. doi: 10.1002/cpa.20253. Google Scholar [34] A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbb R^N$,, Math. Rep., 2 (1984). Google Scholar [35] S. Kaplan, On the growth of solutions of quasilinear parabolic equations,, Comm. Pure Appl. Math., 16 (1963), 305. doi: 10.1002/cpa.3160160307. Google Scholar [36] M. Koslowski, A. M. Cuitino and M. Ortiz, A phasefield theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystal,, J. Mech. Phys. Solids, 50 (2002), 2597. doi: 10.1016/S0022-5096(02)00037-6. Google Scholar [37] H. A. Levine and L. E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time,, J. Differential Equations, 16 (1974), 319. doi: 10.1016/0022-0396(74)90018-7. Google Scholar [38] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I,, Springer-Verlag, (1972). Google Scholar [39] S.-O. Londen and H. Petzeltová, Convergence of solutions of a non-local phase-field system,, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 653. doi: 10.3934/dcdss.2011.4.653. Google Scholar [40] A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations,, Arch. Ration. Mech. Anal., 199 (2011), 493. doi: 10.1007/s00205-010-0354-2. Google Scholar [41] Y. Nec, A. A. Nepomnyashchy and A. A. Golovin, Front-type solutions of fractional Allen-Cahn equation,, Phys. D, 237 (2008), 3237. doi: 10.1016/j.physd.2008.08.002. Google Scholar [42] D. Schertzer, M. Larcheveque, J. Duan, V. V. Yanovsky and S. Lovejoy, Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises,, J. Math. Phys., 42 (2001), 200. doi: 10.1063/1.1318734. Google Scholar [43] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,, Amer. Math. Soc., (1997). Google Scholar [44] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary,, J. Math. Pures Appl., 101 (2014), 275. doi: 10.1016/j.matpur.2013.06.003. Google Scholar [45] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, Discrete Contin. Dyn. Syst., 33 (2013), 2105. Google Scholar [46] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar [47] L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous diffusion: A tutorial,, in Order and Chaos (ed. T. Bountis), (2008). Google Scholar [48] M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets,, Potential Anal., 42 (2015), 499. doi: 10.1007/s11118-014-9443-4. Google Scholar [49] M. Warma, Integration by parts formula and regularity of weak solutions of non-local equations involving the fractional $p$-Laplacian with Neumann and Robin boundary conditions on open sets,, preprint., (). Google Scholar
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