June  2017, 22(4): 1645-1671. doi: 10.3934/dcdsb.2017079

Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

Received  December 2015 Revised  November 2016 Published  February 2017

Fund Project: The author is supported by NSF of China under Grant 11501289

In this paper, we first prove the well-posedness for the nonautonomous reaction-diffusion equations with fractional diffusion in the locally uniform spaces framework. Under very minimal assumptions, then we study the asymptotic behavior of solutions of such equation and show the existence of $(H^{2(\alpha -ε),q}_U(\mathbb{R}^N),H^{2(\alpha -ε),q}_φ(\mathbb{R}^N))(0<ε<\alpha <1)$-uniform(w.r.t.$g∈\mathcal{H}_{L^q_U(\mathbb{R}^N)}(g_0)$) attractor $\mathcal{A}_{\mathcal{H}_{L^q_U(\mathbb{R}^N)}(g_0)}$ with locally uniform external forces being translation uniform bounded but not translation compact in $L_b^p(\mathbb{R};L^q_U(\mathbb{R}^N)).$ The key to that extensions is a new the space-time estimates in locally uniform spaces for the linear fractional power dissipative equation.

Citation: Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079
References:
[1]

F. Abergel, Existence and finite dimensionality of the global attractor for evolution equationds on unbounded domains, J. Differential Equations, 83 (1990), 85-108. doi: 10.1016/0022-0396(90)90070-6. Google Scholar

[2]

B. AndradeA. CarvalhoP. Carvalho-Neto and P. Marín-Rubio, Semilinear fractional differential equations: Global solutions, critical nonlinearities and comparison results, Topological Methods in Nonlinear Analysis, 45 (2015), 439-467. doi: 10.12775/TMNA.2015.022. Google Scholar

[3]

J. ArrietaJ. CholewaT. Dlotko and A. Rodriguez-Bernal, Linear parabolic equations in locally spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293. doi: 10.1142/S0218202504003234. Google Scholar

[4]

J. ArrietaJ. W. CholewaT. Dlotko and A. Rodriguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domain, Nonlinear Anal., 56 (2004), 515-554. doi: 10.1016/j.na.2003.09.023. Google Scholar

[5]

J. Arrieta, N. Moya and A. Rodriguez-Bernal, {Asymptotic behavior of reaction-diffusion equations in weighted Sobolev spaces}, 2009, Submitted.Google Scholar

[6]

A. V. Babin and M. I. Vishik, Attractors of Evolutions North-Holland, Amsterdam, 1992. Google Scholar

[7]

A. Babin and M. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243. doi: 10.1017/S0308210500031498. Google Scholar

[8]

A. Carvalho and T. Dlotko, Partly dissipative systems in locally uniform spaces, Colloq. Math., 100 (2004), 221-242. doi: 10.4064/cm100-2-6. Google Scholar

[9]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems Applied Mathematical Sciences 182, Springer-Verlag, 2013. doi: 10.1007/978-1-4614-4581-4. Google Scholar

[10]

Z. Q. ChenP. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1307-1329. doi: 10.4171/JEMS/231. Google Scholar

[11]

V. Chepyzhov and M. Vishik, {Non-autonomous evolutionary equations with translation compact symbols and their attractors, C. R. Acad. Sci. Paris Sér. I, 321 (1995), 153-158. Google Scholar

[12]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics volume 49 of American Mathematical Society Colloquium Publications, AMS, Providence, RI, 2002. Google Scholar

[13]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333. Google Scholar

[14]

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

[15]

J. Cholewa and T. Dlotko, Cauchy problems in weighted Lebesgue spaces, Czechoslovak Mathematical Journal, 54 (2004), 991-1013. doi: 10.1007/s10587-004-6447-z. Google Scholar

[16]

J. Cholewa and A. Rodriguez-Bernal, Extremal equilibria for dissipative parabolic equations in locally uniform spaces, Math. Model Methods Appl. Sci., 19 (2009), 1995-2037. doi: 10.1142/S0218202509004029. Google Scholar

[17]

J. Cholewa and A. Rodriguez-Bernal, Extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations, J. Differential Equations, 249 (2010), 485-525. doi: 10.1016/j.jde.2010.04.006. Google Scholar

[18]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping Mem. Amer. Math. Soc. , 195 (2008), viii+183 pp. doi: 10.1090/memo/0912. Google Scholar

[19]

T. DlotkoM. Kania and C. Sun, Pseudodifferential parabolic equations in uniform spaces, Applicable Analysis: An International Journal, 93 (2014), 14-34. doi: 10.1080/00036811.2012.753587. Google Scholar

[20]

M. A. Efendiev and S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in an bounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688. doi: 10.1002/cpa.1011. Google Scholar

[21]

J. K. Hale, Asymptotic Behavior of Dissipative Systems Amer. Math. Soc. Providence, RI, 1988. Google Scholar

[22]

A. Haraux, Systemes Dynamiques Dissipatifs et Applications Paris, Masson, 1991. Google Scholar

[23]

D. Henry, Geometric Theory of Semilinear Parabolic Equations Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, 1981. Google Scholar

[24]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740. Google Scholar

[25]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations Leizioni Lincei/Canbridge Univ. Press, Cambridge/New York, 1991. doi: 10.1017/CBO9780511569418. Google Scholar

[26]

X. Li and S. Ruan, Attractors for non-autonomous parabolic problems with singular initial data, J. Differential Equations, 251 (2011), 728-757. doi: 10.1016/j.jde.2011.05.015. Google Scholar

[27]

S. S. LuH. Q. Wu and C. K. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719. doi: 10.3934/dcds.2005.13.701. Google Scholar

[28]

Q. F. MaS. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana University Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. Google Scholar

[29]

V. I. Mazya and T. O. Shaposhnikova, On the Bourgain, Brezis, and Mironescu Theorem Concerning Limiting Embeddings of Fractional Sobolev Spaces, Journal Funct. Anal., 195 (2002), 230-238. doi: 10.1006/jfan.2002.3955. Google Scholar

[30]

C. X. MiaoB. Q. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Analysis, 68 (2008), 461-484. doi: 10.1016/j.na.2006.11.011. Google Scholar

[31]

A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domain sexistence and comparison, Nonlinearity, 8 (1995), 743-768. Google Scholar

[32]

I. MoiseR. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst., 10 (2004), 473-496. doi: 10.3934/dcds.2004.10.473. Google Scholar

[33]

J. Robinson, Infinite-dimensional Dynamical Systems Cambridge University Press Texes in Applied Mathematics, Series, 2002. doi: 10.1007/978-94-010-0732-0. Google Scholar

[34]

C. SunD. Cao and J. Duan, Uniform attractors for non-autonomous wave equations with nonlinear damping, SIAM J. Applied Dynamical Systems, 6 (2007), 293-318. doi: 10.1137/060663805. Google Scholar

[35]

R. Temam, Infinite-dimensional Systems in Mechanics and Physics Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-0645-3. Google Scholar

[36]

J. L. Vázquez, Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators, in Nonlinear elliptic and parabolic differential equations, Disc. Cont. Dyn. Syst. S, 4 (2014), 857-885. Google Scholar

[37]

J. L. Vázquez, Nonlinear Diffusion with Fractional Laplacian Operators, in Nonlinear partial differential equations: the Abel Symposium 2010, Holden, Helge & Karlsen, Kenneth H. eds., Springer, 7 (2012), 271-298. doi: 10.1007/978-3-642-25361-4. Google Scholar

[38]

J. L. Vázquez and B. Volzone, Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type, J. Math. Pures Appl., 101 (2014), 553-582. doi: 10.1016/j.matpur.2013.07.001. Google Scholar

[39]

B. X. Wang, Attractors for reaction-Diffusion equation in unbounded domains, Physica D, 128 (1999), 41-52. doi: 10.1016/S0167-2789(98)00304-2. Google Scholar

[40]

M. H. Yang and C. Y. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity, Trans. Amer. Math. Soc., 361 (2009), 1069-1101. doi: 10.1090/S0002-9947-08-04680-1. Google Scholar

[41]

G. C. Yue and C. K. Zhong, Dynamics of non-autonomous reaction-diffusion equations in locally uniform spaces, Top. Methods Nonlinear Anal., 46 (2015), 935-965. Google Scholar

[42]

G. C. Yue and C. K. Zhong, Global attractors for the Gray-Scott equations in locally uniform spaces, Discrete Contin. Dyn. Syst. B, 21 (2016), 337-356. doi: 10.3934/dcdsb.2016.21.337. Google Scholar

[43]

S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain and Kolmogorov's epsilon-entropy, Math. Nachr., 232 (2001), 129-179. doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.0.CO;2-T. Google Scholar

[44]

S. Zelik, The attractor for a nonlinear hyperbolic equation in the unbounded domain, Discrete Contin. Dyn. Syst., 7 (2001), 593-641. doi: 10.3934/dcds.2001.7.593. Google Scholar

[45]

C. ZhongM. Yang and C. Sun, The existence of global attractors for norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008. Google Scholar

show all references

References:
[1]

F. Abergel, Existence and finite dimensionality of the global attractor for evolution equationds on unbounded domains, J. Differential Equations, 83 (1990), 85-108. doi: 10.1016/0022-0396(90)90070-6. Google Scholar

[2]

B. AndradeA. CarvalhoP. Carvalho-Neto and P. Marín-Rubio, Semilinear fractional differential equations: Global solutions, critical nonlinearities and comparison results, Topological Methods in Nonlinear Analysis, 45 (2015), 439-467. doi: 10.12775/TMNA.2015.022. Google Scholar

[3]

J. ArrietaJ. CholewaT. Dlotko and A. Rodriguez-Bernal, Linear parabolic equations in locally spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293. doi: 10.1142/S0218202504003234. Google Scholar

[4]

J. ArrietaJ. W. CholewaT. Dlotko and A. Rodriguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domain, Nonlinear Anal., 56 (2004), 515-554. doi: 10.1016/j.na.2003.09.023. Google Scholar

[5]

J. Arrieta, N. Moya and A. Rodriguez-Bernal, {Asymptotic behavior of reaction-diffusion equations in weighted Sobolev spaces}, 2009, Submitted.Google Scholar

[6]

A. V. Babin and M. I. Vishik, Attractors of Evolutions North-Holland, Amsterdam, 1992. Google Scholar

[7]

A. Babin and M. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243. doi: 10.1017/S0308210500031498. Google Scholar

[8]

A. Carvalho and T. Dlotko, Partly dissipative systems in locally uniform spaces, Colloq. Math., 100 (2004), 221-242. doi: 10.4064/cm100-2-6. Google Scholar

[9]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems Applied Mathematical Sciences 182, Springer-Verlag, 2013. doi: 10.1007/978-1-4614-4581-4. Google Scholar

[10]

Z. Q. ChenP. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1307-1329. doi: 10.4171/JEMS/231. Google Scholar

[11]

V. Chepyzhov and M. Vishik, {Non-autonomous evolutionary equations with translation compact symbols and their attractors, C. R. Acad. Sci. Paris Sér. I, 321 (1995), 153-158. Google Scholar

[12]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics volume 49 of American Mathematical Society Colloquium Publications, AMS, Providence, RI, 2002. Google Scholar

[13]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333. Google Scholar

[14]

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

[15]

J. Cholewa and T. Dlotko, Cauchy problems in weighted Lebesgue spaces, Czechoslovak Mathematical Journal, 54 (2004), 991-1013. doi: 10.1007/s10587-004-6447-z. Google Scholar

[16]

J. Cholewa and A. Rodriguez-Bernal, Extremal equilibria for dissipative parabolic equations in locally uniform spaces, Math. Model Methods Appl. Sci., 19 (2009), 1995-2037. doi: 10.1142/S0218202509004029. Google Scholar

[17]

J. Cholewa and A. Rodriguez-Bernal, Extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations, J. Differential Equations, 249 (2010), 485-525. doi: 10.1016/j.jde.2010.04.006. Google Scholar

[18]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping Mem. Amer. Math. Soc. , 195 (2008), viii+183 pp. doi: 10.1090/memo/0912. Google Scholar

[19]

T. DlotkoM. Kania and C. Sun, Pseudodifferential parabolic equations in uniform spaces, Applicable Analysis: An International Journal, 93 (2014), 14-34. doi: 10.1080/00036811.2012.753587. Google Scholar

[20]

M. A. Efendiev and S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in an bounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688. doi: 10.1002/cpa.1011. Google Scholar

[21]

J. K. Hale, Asymptotic Behavior of Dissipative Systems Amer. Math. Soc. Providence, RI, 1988. Google Scholar

[22]

A. Haraux, Systemes Dynamiques Dissipatifs et Applications Paris, Masson, 1991. Google Scholar

[23]

D. Henry, Geometric Theory of Semilinear Parabolic Equations Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, 1981. Google Scholar

[24]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740. Google Scholar

[25]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations Leizioni Lincei/Canbridge Univ. Press, Cambridge/New York, 1991. doi: 10.1017/CBO9780511569418. Google Scholar

[26]

X. Li and S. Ruan, Attractors for non-autonomous parabolic problems with singular initial data, J. Differential Equations, 251 (2011), 728-757. doi: 10.1016/j.jde.2011.05.015. Google Scholar

[27]

S. S. LuH. Q. Wu and C. K. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719. doi: 10.3934/dcds.2005.13.701. Google Scholar

[28]

Q. F. MaS. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana University Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. Google Scholar

[29]

V. I. Mazya and T. O. Shaposhnikova, On the Bourgain, Brezis, and Mironescu Theorem Concerning Limiting Embeddings of Fractional Sobolev Spaces, Journal Funct. Anal., 195 (2002), 230-238. doi: 10.1006/jfan.2002.3955. Google Scholar

[30]

C. X. MiaoB. Q. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Analysis, 68 (2008), 461-484. doi: 10.1016/j.na.2006.11.011. Google Scholar

[31]

A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domain sexistence and comparison, Nonlinearity, 8 (1995), 743-768. Google Scholar

[32]

I. MoiseR. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst., 10 (2004), 473-496. doi: 10.3934/dcds.2004.10.473. Google Scholar

[33]

J. Robinson, Infinite-dimensional Dynamical Systems Cambridge University Press Texes in Applied Mathematics, Series, 2002. doi: 10.1007/978-94-010-0732-0. Google Scholar

[34]

C. SunD. Cao and J. Duan, Uniform attractors for non-autonomous wave equations with nonlinear damping, SIAM J. Applied Dynamical Systems, 6 (2007), 293-318. doi: 10.1137/060663805. Google Scholar

[35]

R. Temam, Infinite-dimensional Systems in Mechanics and Physics Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-0645-3. Google Scholar

[36]

J. L. Vázquez, Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators, in Nonlinear elliptic and parabolic differential equations, Disc. Cont. Dyn. Syst. S, 4 (2014), 857-885. Google Scholar

[37]

J. L. Vázquez, Nonlinear Diffusion with Fractional Laplacian Operators, in Nonlinear partial differential equations: the Abel Symposium 2010, Holden, Helge & Karlsen, Kenneth H. eds., Springer, 7 (2012), 271-298. doi: 10.1007/978-3-642-25361-4. Google Scholar

[38]

J. L. Vázquez and B. Volzone, Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type, J. Math. Pures Appl., 101 (2014), 553-582. doi: 10.1016/j.matpur.2013.07.001. Google Scholar

[39]

B. X. Wang, Attractors for reaction-Diffusion equation in unbounded domains, Physica D, 128 (1999), 41-52. doi: 10.1016/S0167-2789(98)00304-2. Google Scholar

[40]

M. H. Yang and C. Y. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity, Trans. Amer. Math. Soc., 361 (2009), 1069-1101. doi: 10.1090/S0002-9947-08-04680-1. Google Scholar

[41]

G. C. Yue and C. K. Zhong, Dynamics of non-autonomous reaction-diffusion equations in locally uniform spaces, Top. Methods Nonlinear Anal., 46 (2015), 935-965. Google Scholar

[42]

G. C. Yue and C. K. Zhong, Global attractors for the Gray-Scott equations in locally uniform spaces, Discrete Contin. Dyn. Syst. B, 21 (2016), 337-356. doi: 10.3934/dcdsb.2016.21.337. Google Scholar

[43]

S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain and Kolmogorov's epsilon-entropy, Math. Nachr., 232 (2001), 129-179. doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.0.CO;2-T. Google Scholar

[44]

S. Zelik, The attractor for a nonlinear hyperbolic equation in the unbounded domain, Discrete Contin. Dyn. Syst., 7 (2001), 593-641. doi: 10.3934/dcds.2001.7.593. Google Scholar

[45]

C. ZhongM. Yang and C. Sun, The existence of global attractors for norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008. Google Scholar

[1]

Gaocheng Yue, Chengkui Zhong. Global attractors for the Gray-Scott equations in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 337-356. doi: 10.3934/dcdsb.2016.21.337

[2]

Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 55-66. doi: 10.3934/dcdss.2009.2.55

[3]

Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343

[4]

Martin Michálek, Dalibor Pražák, Jakub Slavík. Semilinear damped wave equation in locally uniform spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1673-1695. doi: 10.3934/cpaa.2017080

[5]

Gaocheng Yue. Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-22. doi: 10.3934/dcdsb.2019101

[6]

Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581

[7]

Ciprian G. Gal, Mahamadi Warma. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1279-1319. doi: 10.3934/dcds.2016.36.1279

[8]

Yuncheng You, Caidi Zhao, Shengfan Zhou. The existence of uniform attractors for 3D Brinkman-Forchheimer equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3787-3800. doi: 10.3934/dcds.2012.32.3787

[9]

Messoud Efendiev, Alain Miranville. Finite dimensional attractors for reaction-diffusion equations in $R^n$ with a strong nonlinearity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 399-424. doi: 10.3934/dcds.1999.5.399

[10]

P.E. Kloeden, Victor S. Kozyakin. Uniform nonautonomous attractors under discretization. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 423-433. doi: 10.3934/dcds.2004.10.423

[11]

Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875

[12]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891

[13]

Peter E. Kloeden, Thomas Lorenz. Pullback attractors of reaction-diffusion inclusions with space-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1909-1964. doi: 10.3934/dcdsb.2017114

[14]

Yuncheng You. Random attractors and robustness for stochastic reversible reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 301-333. doi: 10.3934/dcds.2014.34.301

[15]

Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43

[16]

Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281

[17]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-21. doi: 10.3934/dcdss.2020083

[18]

María Anguiano, Tomás Caraballo, José Real, José Valero. Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 307-326. doi: 10.3934/dcdsb.2010.14.307

[19]

Kazuo Yamazaki, Xueying Wang. Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 559-579. doi: 10.3934/mbe.2017033

[20]

Susanna Terracini, Gianmaria Verzini, Alessandro Zilio. Uniform Hölder regularity with small exponent in competition-fractional diffusion systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2669-2691. doi: 10.3934/dcds.2014.34.2669

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (12)
  • HTML views (1)
  • Cited by (0)

Other articles
by authors

[Back to Top]