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

Ciprian G. Gal; Mahamadi Warma

doi:10.3934/dcds.2016.36.1279

Previous Article
Wandering continua for rational maps
DCDS Home
This Issue
Next Article
Rigidity of Hamenstädt metrics of Anosov flows

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
Related Papers

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 & 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002. Google Scholar
[13]	J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000. doi: 10.1017/CBO9780511526404. Google Scholar
[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. Google Scholar
[15]	E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158. Google Scholar
[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. Google Scholar
[17]	S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions,, , (). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	C. G. Gal and M. Warma, Long-term behavior of reaction-diffusion equations with nonlocal boundary conditions on rough domains,, submitted., (). Google Scholar
[25]	C. G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractionalp-Laplacian,, submitted., (). Google Scholar
[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. Google Scholar
[27]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman, Boston, MA, 1985. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	Q. Y. Guan and Z. M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424. doi: 10.1142/S021949370500150X. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbb R^N$, Math. Rep., 2 (1984), xiv+221 pp. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[43]	R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1997. Google Scholar
[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. Google Scholar
[45]	R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. Google Scholar
[46]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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., (). Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002. Google Scholar
[13]	J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000. doi: 10.1017/CBO9780511526404. Google Scholar
[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. Google Scholar
[15]	E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158. Google Scholar
[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. Google Scholar
[17]	S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions,, , (). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	C. G. Gal and M. Warma, Long-term behavior of reaction-diffusion equations with nonlocal boundary conditions on rough domains,, submitted., (). Google Scholar
[25]	C. G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractionalp-Laplacian,, submitted., (). Google Scholar
[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. Google Scholar
[27]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman, Boston, MA, 1985. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	Q. Y. Guan and Z. M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424. doi: 10.1142/S021949370500150X. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbb R^N$, Math. Rep., 2 (1984), xiv+221 pp. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[43]	R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1997. Google Scholar
[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. Google Scholar
[45]	R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. Google Scholar
[46]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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., (). Google Scholar

[1]	Eugenio Montefusco, Benedetta Pellacci, Gianmaria Verzini. Fractional diffusion with Neumann boundary conditions: The logistic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2175-2202. doi: 10.3934/dcdsb.2013.18.2175
[2]	Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083
[3]	Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1
[4]	Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067
[5]	Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39
[6]	Keng Deng. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5945-5957. doi: 10.3934/dcdsb.2019114
[7]	Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083
[8]	Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581
[9]	Antonio Suárez. A logistic equation with degenerate diffusion and Robin boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1255-1267. doi: 10.3934/cpaa.2008.7.1255
[10]	Jean-Daniel Djida, Arran Fernandez, Iván Area. Well-posedness results for fractional semi-linear wave equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 569-597. doi: 10.3934/dcdsb.2019255
[11]	Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020
[12]	Bo Li, Xiaoyan Zhang. Steady states of a Sel'kov-Schnakenberg reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1009-1023. doi: 10.3934/dcdss.2017053
[13]	Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019
[14]	Narcisa Apreutesei, Vitaly Volpert. Reaction-diffusion waves with nonlinear boundary conditions. Networks & Heterogeneous Media, 2013, 8 (1) : 23-35. doi: 10.3934/nhm.2013.8.23
[15]	Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155
[16]	Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic & Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039
[17]	Zhenhua Zhang. Asymptotic behavior of solutions to the phase-field equations with neumann boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 683-693. doi: 10.3934/cpaa.2005.4.683
[18]	Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051
[19]	Costică Moroşanu. Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1567-1587. doi: 10.3934/dcdss.2020089
[20]	Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319

2019 Impact Factor: 1.338

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors