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March  2018, 23(2): 749-763. doi: 10.3934/dcdsb.2018041

Long-time behavior of a class of nonlocal partial differential equations

Chang Zhang 1, , Fang Li 2,, and  Jinqiao Duan 3,

1. 

School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China
2. 

School of Mathematics and Statistics, Xidian University, Xi'an 710126, China
3. 

Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

* Corresponding author

Received  October 2016 Revised  October 2017 Published  March 2018 Early access  December 2017

Fund Project: Zhang was supported by NSFC Grant (11701230).

This work is devoted to investigate the well-posedness and long-time behavior of solutions for the following nonlocal nonlinear partial differential equations in a bounded domain
$\begin{align*}u_t+(-Δ)^{σ/2}u +f(u) = g.\end{align*}$
Firstly, due to the lack of an upper growth restriction of the nonlinearity $f$, we have to utilize a weak compactness approach in an Orlicz space to obtain the well-posedness of weak solutions for the equations. We then establish the existence of
$(L^2_0(Ω), L^2_0(Ω))$
-absorbing sets and
$(L^2_0(Ω), H^{σ/2}_0(Ω))$
-absorbing sets for the solution semigroup
$\{S(t)\}_{t≥q 0}$
. Finally, we prove the existence of
$(L^2_0(Ω), L^2_0(Ω))$
-global attractor and
$(L^2_0(Ω), H^{σ/2}_0(Ω))$
-global attractor by a asymptotic compactness method.
Keywords: Fractional Laplacian, long-time behavior, weak solution, nonlocal partial differential equation, bounded domain.
Mathematics Subject Classification: Primary: 35R11, 35D30, 35K61; Secondary: 35A01, 35B41.
Citation: Chang Zhang, Fang Li, Jinqiao Duan. Long-time behavior of a class of nonlocal partial differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 749-763. doi: 10.3934/dcdsb.2018041
References:
[1]

D. Applebaum, Lévy Processes and Stochastic Calculus, Second edition. Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009.

[2]

I. Athanasopoulos and L. A. Caffarelli, Continuity of the temperature in boundary heat control problems, Adv. Math., 224 (2010), 293-315.  doi: 10.1016/j.aim.2009.11.010.

[3]

A. Babin and M. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[4]

J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996.

[5]

C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Math. Roy. Soc. Edinb., 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.

[6]

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

[7]

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

[8]

Z. Chen and R. Song, Hardy inequality for censored stable processes, Tohoku Math. J., 55 (2003), 439-450.  doi: 10.2748/tmj/1113247482.

[9]

R. Cont and P. Tankov, Financial Modelling With Jump Processes, Boca Raton, FL: Chapman Hall/CRC, 2004.

[10] J. Duan, An Introduction to Stochastic Dynamics, Cambridge University Press, New York, 2015. 
[11]

X. Fernández-Real and X. Ros-Oton, Boundary regularity for the fractional heat equation, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 10 (2016), 49-64.  doi: 10.1007/s13398-015-0218-6.

[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. Walter de Gruyter & Co., Berlin, 2011.

[13]

P. Geredeli and A. Khanmamedov, Long-time dynamics of the parabolic $p$-Laplacian equation, Commun. Pure Appl. Anal., 12 (2013), 735-754. 

[14]

A. Kiselev, F. 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.

[15]

M. Krasnoselskii and Y. Rutickii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961.

[16]

J. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, New York: Springer-Verlag, Vol Ⅰ, 1973.

[17]

H. Lu, P. Bates, S. Lü and M. Zhang, Dynamics of the 3-D fractional complex GinzburgLandau equation, J. Differ. Equ., 259 (2015), 5276-5301.  doi: 10.1016/j.jde.2015.06.028.

[18]

H. Lu, P. Bates, S. Lü and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295.  doi: 10.4310/CMS.2016.v14.n1.a11.

[19]

J. Mercado, E. Guido, A. Sánchez-Sesma, M. ͘ñiguez and A. González, Analysis of the Blasius Formula and the Navier-Stokes Fractional Equation, Chapter Fluid Dynamics in Physics, Engineering and Environmental Applications Part of the series Environmental Science and Engineering, (2012), 475–480.

[20]

R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A: Mathematical and General, 37 (2004), 161-208.  doi: 10.1088/0305-4470/37/31/R01.

[21]

E. 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.

[22]

A. de Pablo, F. Quirós, A. Rodriguez and J. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.  doi: 10.1016/j.aim.2010.07.017.

[23]

A. de Pablo, F. Quirós, A. Rodriguez and J. Vázquez, A general fractional porous medium equation, Comm. Pure Applied Math., 65 (2012), 1242-1284.  doi: 10.1002/cpa.21408.

[24]

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.

[25]

J. Simon, Compact sets in the space Lp(O, T; B), Annali di Matematica Pura ed Applicata, 146 (1987), 65-96. 

[26]

R. Song and Z. Vondraček, Potential theory of subordinate killed Brownian motion in a domain, Probab. Theory Relat. Fields, 125 (2003), 578-592.  doi: 10.1007/s00440-002-0251-1.

[27]

P. Stinga and J. Torrea, Extension problem and Harnack's inequality for some fractional operators, Commun. Partial Differ. Equ., 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.

[28]

J. 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.

[29]

M. Yang, C. Sun and C. Zhong, Global attractors for p-Laplacian equation, J. Math. Anal. Appl., 327 (2007), 1130-1142.  doi: 10.1016/j.jmaa.2006.04.085.

[30]

X. Zhang, Stochastic lagrangian particle approach to fractal Navier-Stokes equations, Commun. Math. Phys., 311 (2012), 133-155.  doi: 10.1007/s00220-012-1414-2.

[31]

C. Zhang, J. Zhang and C. Zhong, Existence of weak solutions for fractional porous medium equations with nonlinear term, Appl. Math. Lett., 61 (2016), 95-101.  doi: 10.1016/j.aml.2016.05.001.

[32]

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

show all references

References:
[1]

D. Applebaum, Lévy Processes and Stochastic Calculus, Second edition. Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009.

[2]

I. Athanasopoulos and L. A. Caffarelli, Continuity of the temperature in boundary heat control problems, Adv. Math., 224 (2010), 293-315.  doi: 10.1016/j.aim.2009.11.010.

[3]

A. Babin and M. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[4]

J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996.

[5]

C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Math. Roy. Soc. Edinb., 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.

[6]

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

[7]

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

[8]

Z. Chen and R. Song, Hardy inequality for censored stable processes, Tohoku Math. J., 55 (2003), 439-450.  doi: 10.2748/tmj/1113247482.

[9]

R. Cont and P. Tankov, Financial Modelling With Jump Processes, Boca Raton, FL: Chapman Hall/CRC, 2004.

[10] J. Duan, An Introduction to Stochastic Dynamics, Cambridge University Press, New York, 2015. 
[11]

X. Fernández-Real and X. Ros-Oton, Boundary regularity for the fractional heat equation, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 10 (2016), 49-64.  doi: 10.1007/s13398-015-0218-6.

[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. Walter de Gruyter & Co., Berlin, 2011.

[13]

P. Geredeli and A. Khanmamedov, Long-time dynamics of the parabolic $p$-Laplacian equation, Commun. Pure Appl. Anal., 12 (2013), 735-754. 

[14]

A. Kiselev, F. 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.

[15]

M. Krasnoselskii and Y. Rutickii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961.

[16]

J. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, New York: Springer-Verlag, Vol Ⅰ, 1973.

[17]

H. Lu, P. Bates, S. Lü and M. Zhang, Dynamics of the 3-D fractional complex GinzburgLandau equation, J. Differ. Equ., 259 (2015), 5276-5301.  doi: 10.1016/j.jde.2015.06.028.

[18]

H. Lu, P. Bates, S. Lü and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295.  doi: 10.4310/CMS.2016.v14.n1.a11.

[19]

J. Mercado, E. Guido, A. Sánchez-Sesma, M. ͘ñiguez and A. González, Analysis of the Blasius Formula and the Navier-Stokes Fractional Equation, Chapter Fluid Dynamics in Physics, Engineering and Environmental Applications Part of the series Environmental Science and Engineering, (2012), 475–480.

[20]

R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A: Mathematical and General, 37 (2004), 161-208.  doi: 10.1088/0305-4470/37/31/R01.

[21]

E. 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.

[22]

A. de Pablo, F. Quirós, A. Rodriguez and J. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.  doi: 10.1016/j.aim.2010.07.017.

[23]

A. de Pablo, F. Quirós, A. Rodriguez and J. Vázquez, A general fractional porous medium equation, Comm. Pure Applied Math., 65 (2012), 1242-1284.  doi: 10.1002/cpa.21408.

[24]

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.

[25]

J. Simon, Compact sets in the space Lp(O, T; B), Annali di Matematica Pura ed Applicata, 146 (1987), 65-96. 

[26]

R. Song and Z. Vondraček, Potential theory of subordinate killed Brownian motion in a domain, Probab. Theory Relat. Fields, 125 (2003), 578-592.  doi: 10.1007/s00440-002-0251-1.

[27]

P. Stinga and J. Torrea, Extension problem and Harnack's inequality for some fractional operators, Commun. Partial Differ. Equ., 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.

[28]

J. 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.

[29]

M. Yang, C. Sun and C. Zhong, Global attractors for p-Laplacian equation, J. Math. Anal. Appl., 327 (2007), 1130-1142.  doi: 10.1016/j.jmaa.2006.04.085.

[30]

X. Zhang, Stochastic lagrangian particle approach to fractal Navier-Stokes equations, Commun. Math. Phys., 311 (2012), 133-155.  doi: 10.1007/s00220-012-1414-2.

[31]

C. Zhang, J. Zhang and C. Zhong, Existence of weak solutions for fractional porous medium equations with nonlinear term, Appl. Math. Lett., 61 (2016), 95-101.  doi: 10.1016/j.aml.2016.05.001.

[32]

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

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