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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

 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.
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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