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

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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

Ciprian G. Gal ^1, and Mahamadi Warma ^2,

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

Abstract
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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.

Keywords: global attractor, asymptotic behavior., convergence to steady states, The fractional Laplace operator, semi-linear reaction-diffusion equation, fractional Neumann and Robin boundary conditions on open sets.

Mathematics Subject Classification: 35R11, 35A15, 35B41, 35K6.

Citation: Ciprian G. Gal, Mahamadi Warma. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete and Continuous Dynamical Systems, 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), 403-407.
[2]	D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften, 314, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03282-4.
[3]	N. D. Alikakos, $L^p$-bounds of solutions to reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.
[4]	F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010. doi: 10.1090/surv/165.
[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), 399-408. doi: 10.2140/pjm.1959.9.399.
[6]	K. Bogdan, K. Burdzy and Z. Q. Chen, Censored stable processes, Probab. Theory Related Fields, 127 (2003), 89-152. doi: 10.1007/s00440-003-0275-1.
[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), 77-101. doi: 10.1007/BF02868470.
[8]	L. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.
[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), 425-461. doi: 10.1007/s00222-007-0086-6.
[10]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[11]	Z.-Q. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on $d$-sets, Stochastic Process. Appl., 108 (2003), 27-62. doi: 10.1016/S0304-4149(03)00105-4.
[12]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002.
[13]	J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000. doi: 10.1017/CBO9780511526404.
[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), x+119 pp. doi: 10.1090/memo/0857.
[15]	E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158.
[16]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[17]	S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions,, , ().
[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), 493-540. doi: 10.1142/S0218202512500546.
[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), 1-21. doi: 10.1016/j.jde.2003.10.026.
[20]	H. Gajewski and J. Griepentrog, A descent method for the free energy of multicomponent systems, Discrete Contin. Dyn. Syst., 15 (2006), 505-528. doi: 10.3934/dcds.2006.15.505.
[21]	C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, J. Differential Equations, 253 (2012), 126-166. doi: 10.1016/j.jde.2012.02.010.
[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), 85-106. doi: 10.1007/s00332-011-9109-y.
[23]	C. G. Gal and M. Grasselli, Longtime behavior of nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 34 (2014), 145-179. doi: 10.3934/dcds.2014.34.145.
[24]	C. G. Gal and M. Warma, Long-term behavior of reaction-diffusion equations with nonlocal boundary conditions on rough domains,, submitted., ().
[25]	C. G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractionalp-Laplacian,, submitted., ().
[26]	A. Garroni and S. Müller, A variational model for dislocations in the line tension limit, Arch. Ration. Mech. Anal., 181 (2006), 535-578. doi: 10.1007/s00205-006-0432-7.
[27]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman, Boston, MA, 1985.
[28]	Q. Y. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329. doi: 10.1007/s00220-006-0054-9.
[29]	Q. Y. Guan and Z. M. Ma, Reflected symmetric $\alpha $-stable processes and regional fractional Laplacian, Probab. Theory Related Fields, 134 (2006), 649-694. doi: 10.1007/s00440-005-0438-3.
[30]	Q. Y. Guan and Z. M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424. doi: 10.1142/S021949370500150X.
[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,, in press, (2014). doi: 10.1016/j.anihpc.2014.03.005.
[32]	M. Gunzburger and R. B. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581-1598. doi: 10.1137/090766607.
[33]	M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214. doi: 10.1002/cpa.20253.
[34]	A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbb R^N$, Math. Rep., 2 (1984), xiv+221 pp.
[35]	S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330. doi: 10.1002/cpa.3160160307.
[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), 2597-2635. doi: 10.1016/S0022-5096(02)00037-6.
[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), 319-334. doi: 10.1016/0022-0396(74)90018-7.
[38]	J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972.
[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), 653-670. doi: 10.3934/dcdss.2011.4.653.
[40]	A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2.
[41]	Y. Nec, A. A. Nepomnyashchy and A. A. Golovin, Front-type solutions of fractional Allen-Cahn equation, Phys. D, 237 (2008), 3237-3251. doi: 10.1016/j.physd.2008.08.002.
[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), 200-212. doi: 10.1063/1.1318734.
[43]	R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1997.
[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), 275-302. doi: 10.1016/j.matpur.2013.06.003.
[45]	R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
[46]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[47]	L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous diffusion: A tutorial, in Order and Chaos (ed. T. Bountis), Vol. 10, Patras University Press, 2008.
[48]	M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal., 42 (2015), 499-547. doi: 10.1007/s11118-014-9443-4.
[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., ().

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), 403-407.
[2]	D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften, 314, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03282-4.
[3]	N. D. Alikakos, $L^p$-bounds of solutions to reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.
[4]	F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010. doi: 10.1090/surv/165.
[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), 399-408. doi: 10.2140/pjm.1959.9.399.
[6]	K. Bogdan, K. Burdzy and Z. Q. Chen, Censored stable processes, Probab. Theory Related Fields, 127 (2003), 89-152. doi: 10.1007/s00440-003-0275-1.
[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), 77-101. doi: 10.1007/BF02868470.
[8]	L. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.
[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), 425-461. doi: 10.1007/s00222-007-0086-6.
[10]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[11]	Z.-Q. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on $d$-sets, Stochastic Process. Appl., 108 (2003), 27-62. doi: 10.1016/S0304-4149(03)00105-4.
[12]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002.
[13]	J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000. doi: 10.1017/CBO9780511526404.
[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), x+119 pp. doi: 10.1090/memo/0857.
[15]	E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158.
[16]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[17]	S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions,, , ().
[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), 493-540. doi: 10.1142/S0218202512500546.
[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), 1-21. doi: 10.1016/j.jde.2003.10.026.
[20]	H. Gajewski and J. Griepentrog, A descent method for the free energy of multicomponent systems, Discrete Contin. Dyn. Syst., 15 (2006), 505-528. doi: 10.3934/dcds.2006.15.505.
[21]	C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, J. Differential Equations, 253 (2012), 126-166. doi: 10.1016/j.jde.2012.02.010.
[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), 85-106. doi: 10.1007/s00332-011-9109-y.
[23]	C. G. Gal and M. Grasselli, Longtime behavior of nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 34 (2014), 145-179. doi: 10.3934/dcds.2014.34.145.
[24]	C. G. Gal and M. Warma, Long-term behavior of reaction-diffusion equations with nonlocal boundary conditions on rough domains,, submitted., ().
[25]	C. G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractionalp-Laplacian,, submitted., ().
[26]	A. Garroni and S. Müller, A variational model for dislocations in the line tension limit, Arch. Ration. Mech. Anal., 181 (2006), 535-578. doi: 10.1007/s00205-006-0432-7.
[27]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman, Boston, MA, 1985.
[28]	Q. Y. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329. doi: 10.1007/s00220-006-0054-9.
[29]	Q. Y. Guan and Z. M. Ma, Reflected symmetric $\alpha $-stable processes and regional fractional Laplacian, Probab. Theory Related Fields, 134 (2006), 649-694. doi: 10.1007/s00440-005-0438-3.
[30]	Q. Y. Guan and Z. M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424. doi: 10.1142/S021949370500150X.
[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,, in press, (2014). doi: 10.1016/j.anihpc.2014.03.005.
[32]	M. Gunzburger and R. B. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581-1598. doi: 10.1137/090766607.
[33]	M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214. doi: 10.1002/cpa.20253.
[34]	A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbb R^N$, Math. Rep., 2 (1984), xiv+221 pp.
[35]	S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330. doi: 10.1002/cpa.3160160307.
[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), 2597-2635. doi: 10.1016/S0022-5096(02)00037-6.
[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), 319-334. doi: 10.1016/0022-0396(74)90018-7.
[38]	J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972.
[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), 653-670. doi: 10.3934/dcdss.2011.4.653.
[40]	A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2.
[41]	Y. Nec, A. A. Nepomnyashchy and A. A. Golovin, Front-type solutions of fractional Allen-Cahn equation, Phys. D, 237 (2008), 3237-3251. doi: 10.1016/j.physd.2008.08.002.
[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), 200-212. doi: 10.1063/1.1318734.
[43]	R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1997.
[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), 275-302. doi: 10.1016/j.matpur.2013.06.003.
[45]	R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
[46]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[47]	L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous diffusion: A tutorial, in Order and Chaos (ed. T. Bountis), Vol. 10, Patras University Press, 2008.
[48]	M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal., 42 (2015), 499-547. doi: 10.1007/s11118-014-9443-4.
[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., ().

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