December  2017, 37(12): 6035-6067. doi: 10.3934/dcds.2017260

Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion

1. 

School of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, Guangdong, China

2. 

Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education, and Guangdong Province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, Guangdong, China

3. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, Gansu, China

* Corresponding author: Wen Tan

Received  July 2016 Revised  July 2017 Published  August 2017

Fund Project: The authors were supported by NSF of China (Grant No. 11471148, Grant No. 11522109).

In this paper, we study the dynamics of a non-autonomous reaction diffusion model with the fractional diffusion on the whole space. We firstly prove the existence of a $(L^2,L^2)$ pullback $\mathscr{D}_μ$ -attractor of this model. Then we show that the pullback $\mathscr{D}_μ$ -attractor attract the $\mathscr{D}_μ$ class (especially all $L^2$ -bounded set) in $L^{2+δ}$-norm for any $δ∈[0,∞)$. Moreover, the solution of the model is shown to be continuous in $H^s$ with respect to initial data under a slightly stronger condition on external forcing term. As an application, we prove that the $(L^2,L^2)$ pullback $\mathscr{D}_{μ}$-attractor indeed attract the class of $\mathscr{D}_{μ}$ in $H^s$ -norm, and thus the existence of a $(L^2, H^s)$ pullback $\mathscr{D}_μ$ -attractor is obtained.

Citation: Wen Tan, Chunyou Sun. Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6035-6067. doi: 10.3934/dcds.2017260
References:
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R. A. Adams, Sobolev Spaces, New York: Academic Press, 1975.

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M. AnguianoP. Marín-Rubio and J. Real, Pullback attractors for non-autonomous reaction-diffusion equations with dynamical boundary conditions, J. Math. Anal. Appl., 383 (2011), 608-618.  doi: 10.1016/j.jmaa.2011.05.046.

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H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

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J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.

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C. BardosP. PenelU. Frisch and P.-L. Sulem, Modified dissipativity for a nonlinear evolution equations arising in turbulence, Arch. Rational Mech. Anal., 71 (1979), 237-256.  doi: 10.1007/BF00280598.

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P. BilerC. Imbert and G. Karch, Barenblatt profiles for a nonlocal porous medium equation, C. R. Math. Acad. Sci. Paris., 349 (2011), 641-645.  doi: 10.1016/j.crma.2011.06.003.

[8]

M. BonforteY. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725.

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J. BourgainH. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s\to \text{1}$ and applications, J. Anal. Math., 87 (2002), 77-101.  doi: 10.1007/BF02868470.

[10]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[11]

L. A. CaffarelliF. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow, J. Eur. Math. Soc., 15 (2013), 1701-1746.  doi: 10.4171/JEMS/401.

[12]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.

[13]

D. CaoC. Sun and M. Yang, Dynamical for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872.  doi: 10.1016/j.jde.2015.02.020.

[14]

T. CaraballoG. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[15]

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

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V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002.

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

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P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Lecture Notes in Mathematics, vol. 1871, Springer-Verlag, Berlin, 2006, 1–43. doi: 10.1007/11545989_1.

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H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[20]

E. Di NezzaG. 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.

[21]

T. DlotkoM. Kania and C. Sun, Pseudodifferential parabolic equations; two examples, Topol. Methods Nonlinear Anal., 43 (2014), 463-492.  doi: 10.12775/TMNA.2014.028.

[22]

J. Droniou and C. Imbert, Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182 (2006), 299-331.  doi: 10.1007/s00205-006-0429-2.

[23]

N. Jacob, Pseudo Differential Operators and Markov Processes, Vol. Ⅰ, Ⅱ, Ⅲ, Imperial College Press, London, 2005. doi: 10.1142/9781860947155.

[24]

S. Jarohs and T. Weth, Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 2581-2615.  doi: 10.3934/dcds.2014.34.2581.

[25]

G. Karch, Nonlinear evolution equations with anomalous diffusion, Qualitative Properties of Solutions to Partial Differential Equations, J. Nečas Center for Mathematical Modeling, Charles University, Prague, 5 (2009), 25–68.

[26]

A. KiselevF. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.  doi: 10.1007/s00222-006-0020-3.

[27]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176, 2011. doi: 10.1090/surv/176.

[28]

G. Lukaszewicz, On pullback attractors in $H_0^1$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos, 20 (2010), 2637-2644.  doi: 10.1142/S0218127410027258.

[29]

G. Lukaszewicz, On pullback attractors in $L^p$ for nonautonomous reaction-diffusion equations, Nonlinear Anal., 73 (2010), 350-357.  doi: 10.1016/j.na.2010.03.023.

[30]

V. I. Mazya and T. O. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.  doi: 10.1006/jfan.2002.3955.

[31]

A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.  doi: 10.1016/j.aim.2010.07.017.

[32]

J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0.

[33]

C. Sun and Y. Yuan, $L^p$ -type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052.  doi: 10.1017/S0308210515000177.

[34]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[35]

H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78. Birkhuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[36]

T. Trujillo and B. Wang, Continuity of strong solutions of reaction-diffusion equation in initial data, Nonlinear Anal., 69 (2008), 2525-2532.  doi: 10.1016/j.na.2007.08.032.

[37]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44. 

[38]

J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators, in Nonlinear partial differential equations: the Abel Symposium 2010 (ed. H. Kenneth), Holden, Helge & Karlsen, Springer, 2012,271–298. doi: 10.1007/978-3-642-25361-4_15.

[39]

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.

[40]

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

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, New York: Academic Press, 1975.

[2]

M. AnguianoP. Marín-Rubio and J. Real, Pullback attractors for non-autonomous reaction-diffusion equations with dynamical boundary conditions, J. Math. Anal. Appl., 383 (2011), 608-618.  doi: 10.1016/j.jmaa.2011.05.046.

[3]

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.

[4]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[5]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.

[6]

C. BardosP. PenelU. Frisch and P.-L. Sulem, Modified dissipativity for a nonlinear evolution equations arising in turbulence, Arch. Rational Mech. Anal., 71 (1979), 237-256.  doi: 10.1007/BF00280598.

[7]

P. BilerC. Imbert and G. Karch, Barenblatt profiles for a nonlocal porous medium equation, C. R. Math. Acad. Sci. Paris., 349 (2011), 641-645.  doi: 10.1016/j.crma.2011.06.003.

[8]

M. BonforteY. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725.

[9]

J. BourgainH. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s\to \text{1}$ and applications, J. Anal. Math., 87 (2002), 77-101.  doi: 10.1007/BF02868470.

[10]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[11]

L. A. CaffarelliF. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow, J. Eur. Math. Soc., 15 (2013), 1701-1746.  doi: 10.4171/JEMS/401.

[12]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.

[13]

D. CaoC. Sun and M. Yang, Dynamical for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872.  doi: 10.1016/j.jde.2015.02.020.

[14]

T. CaraballoG. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[15]

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

[16]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002.

[17]

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

[18]

P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Lecture Notes in Mathematics, vol. 1871, Springer-Verlag, Berlin, 2006, 1–43. doi: 10.1007/11545989_1.

[19]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[20]

E. Di NezzaG. 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.

[21]

T. DlotkoM. Kania and C. Sun, Pseudodifferential parabolic equations; two examples, Topol. Methods Nonlinear Anal., 43 (2014), 463-492.  doi: 10.12775/TMNA.2014.028.

[22]

J. Droniou and C. Imbert, Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182 (2006), 299-331.  doi: 10.1007/s00205-006-0429-2.

[23]

N. Jacob, Pseudo Differential Operators and Markov Processes, Vol. Ⅰ, Ⅱ, Ⅲ, Imperial College Press, London, 2005. doi: 10.1142/9781860947155.

[24]

S. Jarohs and T. Weth, Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 2581-2615.  doi: 10.3934/dcds.2014.34.2581.

[25]

G. Karch, Nonlinear evolution equations with anomalous diffusion, Qualitative Properties of Solutions to Partial Differential Equations, J. Nečas Center for Mathematical Modeling, Charles University, Prague, 5 (2009), 25–68.

[26]

A. KiselevF. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.  doi: 10.1007/s00222-006-0020-3.

[27]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176, 2011. doi: 10.1090/surv/176.

[28]

G. Lukaszewicz, On pullback attractors in $H_0^1$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos, 20 (2010), 2637-2644.  doi: 10.1142/S0218127410027258.

[29]

G. Lukaszewicz, On pullback attractors in $L^p$ for nonautonomous reaction-diffusion equations, Nonlinear Anal., 73 (2010), 350-357.  doi: 10.1016/j.na.2010.03.023.

[30]

V. I. Mazya and T. O. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.  doi: 10.1006/jfan.2002.3955.

[31]

A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.  doi: 10.1016/j.aim.2010.07.017.

[32]

J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0.

[33]

C. Sun and Y. Yuan, $L^p$ -type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052.  doi: 10.1017/S0308210515000177.

[34]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[35]

H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78. Birkhuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[36]

T. Trujillo and B. Wang, Continuity of strong solutions of reaction-diffusion equation in initial data, Nonlinear Anal., 69 (2008), 2525-2532.  doi: 10.1016/j.na.2007.08.032.

[37]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44. 

[38]

J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators, in Nonlinear partial differential equations: the Abel Symposium 2010 (ed. H. Kenneth), Holden, Helge & Karlsen, Springer, 2012,271–298. doi: 10.1007/978-3-642-25361-4_15.

[39]

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.

[40]

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

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