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March  2016, 5(1): 61-103. doi: 10.3934/eect.2016.5.61

Elliptic and parabolic equations with fractional diffusion and dynamic 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  July 2015 Revised  November 2015 Published  March 2016

We investigate a class of semilinear parabolic and elliptic problems with fractional dynamic boundary conditions. We introduce two new operators, the so-called fractional Wentzell Laplacian and the fractional Steklov operator, which become essential in our study of these nonlinear problems. Besides giving a complete characterization of well-posedness and regularity of bounded solutions, we also establish the existence of finite-dimensional global attractors and also derive basic conditions for blow-up.
Keywords: elliptic problem, fractional Wentzell boundary conditions, The fractional Laplace operator, fractional Dirichlet-to-Neumann operator., exponential attractor, global attractor, fractional Steklov operator, semilinear reaction-diffusion equation.
Mathematics Subject Classification: Primary: 35J92, 35A15, 35B41; Secondary: 35K6.
Citation: Ciprian G. Gal, Mahamadi Warma. Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions. Evolution Equations & Control Theory, 2016, 5 (1) : 61-103. doi: 10.3934/eect.2016.5.61
References:
[1]

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.  Google Scholar

[2]

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.  Google Scholar

[3]

J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000. doi: 10.1017/CBO9780511526404.  Google Scholar

[4]

D. Daners and P. Drábek, A priori estimates for a class of quasi-linear elliptic equations, Trans. Amer. Math. Soc., 361 (2009), 6475-6500. doi: 10.1090/S0002-9947-09-04839-9.  Google Scholar

[5]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158.  Google Scholar

[6]

A. Di Castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836. doi: 10.1016/j.jfa.2014.05.023.  Google Scholar

[7]

A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers,, Ann. Inst. H. Poincaré Anal. Non Linéaire, ().  doi: 10.1016/j.anihpc.2015.04.003.  Google Scholar

[8]

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.  Google Scholar

[9]

S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions,, Rev. Mat. Iberoam., ().   Google Scholar

[10]

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.  Google Scholar

[11]

M. Efendiev and S. Zelik, Finite-dimensional attractors and exponential attractors for degenerate doubly nonlinear equations, Math. Methods Appl. Sci., 32 (2009), 1638-1668. doi: 10.1002/mma.1102.  Google Scholar

[12]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Second revised and extended edition. De Gruyter Studies in Mathematics, 19. Berlin, 2011.  Google Scholar

[13]

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.  Google Scholar

[14]

C. G. Gal, The role of surface diffusion in dynamic boundary conditions: Where do we stand?, Milan Journal of Mathematics, 83 (2015), 237-278. doi: 10.1007/s00032-015-0242-1.  Google Scholar

[15]

C. G. Gal and M. Warma, Long-term behavior of reaction-diffusion equations with nonlocal boundary conditions on rough domains,, J. Dyn. Diff. Eqns., ().   Google Scholar

[16]

C. G. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Disc. Cont. Dyn. Syst., Series A, 36 (2016), 1279-1319.  Google Scholar

[17]

R. Gorenflo and F. Mainardi, Random walk models approximating symmetric space-fractional diffusion processes, Problems and methods in mathematical physics (Chemnitz, 1999), Oper. Theory Adv. Appl., Birkhäuser, Basel, 121 (2001), 120-145.  Google Scholar

[18]

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.  Google Scholar

[19]

Q. Y. Guan and Z. M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424. doi: 10.1142/S021949370500150X.  Google Scholar

[20]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24. Pitman, Boston, MA, 1985.  Google Scholar

[21]

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.  Google Scholar

[22]

A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbb R^N$, Math. Rep., 2 (1984), xiv+221 pp.  Google Scholar

[23]

M. Kirane, Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type, Hokkaido Math. J., 21 (1992), 221-229. doi: 10.14492/hokmj/1381413677.  Google Scholar

[24]

T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368. doi: 10.1007/s00220-015-2356-2.  Google Scholar

[25]

R. Muralidhar, D. Ramkrishna, H. Nakanishi and D. Jacobs, Anomalous diffusion: A dynamic perspective, Physica A: Statistical Mechanics and its Applications, 167 (1990), 539-559. doi: 10.1016/0378-4371(90)90132-C.  Google Scholar

[26]

E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series 31, Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[27]

S. Umarov and R. Gorenflo, On multi-dimensional random walk models approximating symmetric space-fractional diffusion processes, Fract. Calc. Appl. Anal., 8 (2005), 73-88.  Google Scholar

[28]

J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857.  Google Scholar

[29]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous Diffusion: a tutorial, In "Order and chaos", 10th volume, T. Bountis (ed.), Patras University Press, 2008. Google Scholar

[30]

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.  Google Scholar

[31]

M. Warma, A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains, Commun. Pure Appl. Anal., 14 (2015), 2043-2067. doi: 10.3934/cpaa.2015.14.2043.  Google Scholar

[32]

M. Warma, The fractional Neumann and Robin boundary conditions for the regional fractional $p$-Laplacian, NoDEA Nonlinear Differential Equations Appl., 23 (2016), p1. doi: 10.1007/s00030-016-0354-5.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

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.  Google Scholar

[3]

J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000. doi: 10.1017/CBO9780511526404.  Google Scholar

[4]

D. Daners and P. Drábek, A priori estimates for a class of quasi-linear elliptic equations, Trans. Amer. Math. Soc., 361 (2009), 6475-6500. doi: 10.1090/S0002-9947-09-04839-9.  Google Scholar

[5]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158.  Google Scholar

[6]

A. Di Castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836. doi: 10.1016/j.jfa.2014.05.023.  Google Scholar

[7]

A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers,, Ann. Inst. H. Poincaré Anal. Non Linéaire, ().  doi: 10.1016/j.anihpc.2015.04.003.  Google Scholar

[8]

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.  Google Scholar

[9]

S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions,, Rev. Mat. Iberoam., ().   Google Scholar

[10]

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.  Google Scholar

[11]

M. Efendiev and S. Zelik, Finite-dimensional attractors and exponential attractors for degenerate doubly nonlinear equations, Math. Methods Appl. Sci., 32 (2009), 1638-1668. doi: 10.1002/mma.1102.  Google Scholar

[12]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Second revised and extended edition. De Gruyter Studies in Mathematics, 19. Berlin, 2011.  Google Scholar

[13]

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.  Google Scholar

[14]

C. G. Gal, The role of surface diffusion in dynamic boundary conditions: Where do we stand?, Milan Journal of Mathematics, 83 (2015), 237-278. doi: 10.1007/s00032-015-0242-1.  Google Scholar

[15]

C. G. Gal and M. Warma, Long-term behavior of reaction-diffusion equations with nonlocal boundary conditions on rough domains,, J. Dyn. Diff. Eqns., ().   Google Scholar

[16]

C. G. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Disc. Cont. Dyn. Syst., Series A, 36 (2016), 1279-1319.  Google Scholar

[17]

R. Gorenflo and F. Mainardi, Random walk models approximating symmetric space-fractional diffusion processes, Problems and methods in mathematical physics (Chemnitz, 1999), Oper. Theory Adv. Appl., Birkhäuser, Basel, 121 (2001), 120-145.  Google Scholar

[18]

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.  Google Scholar

[19]

Q. Y. Guan and Z. M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424. doi: 10.1142/S021949370500150X.  Google Scholar

[20]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24. Pitman, Boston, MA, 1985.  Google Scholar

[21]

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.  Google Scholar

[22]

A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbb R^N$, Math. Rep., 2 (1984), xiv+221 pp.  Google Scholar

[23]

M. Kirane, Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type, Hokkaido Math. J., 21 (1992), 221-229. doi: 10.14492/hokmj/1381413677.  Google Scholar

[24]

T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368. doi: 10.1007/s00220-015-2356-2.  Google Scholar

[25]

R. Muralidhar, D. Ramkrishna, H. Nakanishi and D. Jacobs, Anomalous diffusion: A dynamic perspective, Physica A: Statistical Mechanics and its Applications, 167 (1990), 539-559. doi: 10.1016/0378-4371(90)90132-C.  Google Scholar

[26]

E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series 31, Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[27]

S. Umarov and R. Gorenflo, On multi-dimensional random walk models approximating symmetric space-fractional diffusion processes, Fract. Calc. Appl. Anal., 8 (2005), 73-88.  Google Scholar

[28]

J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857.  Google Scholar

[29]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous Diffusion: a tutorial, In "Order and chaos", 10th volume, T. Bountis (ed.), Patras University Press, 2008. Google Scholar

[30]

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.  Google Scholar

[31]

M. Warma, A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains, Commun. Pure Appl. Anal., 14 (2015), 2043-2067. doi: 10.3934/cpaa.2015.14.2043.  Google Scholar

[32]

M. Warma, The fractional Neumann and Robin boundary conditions for the regional fractional $p$-Laplacian, NoDEA Nonlinear Differential Equations Appl., 23 (2016), p1. doi: 10.1007/s00030-016-0354-5.  Google Scholar

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