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 and 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.

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[31]

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

[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.

[33]

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

[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.

[35]

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

[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. 

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[31]

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

[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.

[33]

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

[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.

[35]

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

[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. 

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

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