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 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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