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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 |
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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