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March  2015, 4(1): 39-59. doi: 10.3934/eect.2015.4.39

Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation

1. 

Département de Mathématiques, Université de Batna, Algeria

2. 

CNRS, Laboratoire de Mathématique, Analyse Numérique et EDP, Université de Paris-Sud, F-91405 Orsay Cedex, France

3. 

Laboratoire de Mathématique, Analyse Numérique et EDP, Université de Paris-Sud, F-91405 Orsay Cedex, France

Received  October 2014 Revised  December 2014 Published  February 2015

We consider a nonlocal reaction-diffusion equation with mass conservation, which was originally proposed by Rubinstein and Sternberg as a model for phase separation in a binary mixture. We study the large time behavior of the solution and show that it converges to a stationary solution as $t$ tends to infinity. We also evaluate the rate of convergence. In some special case, we show that the limit solution is constant.
Citation: 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
References:
[1]

R. A. Adams and J. H. F. Fournier, Sobolev Spaces, $2^{nd}$ edition, Academic Press, 2003.  Google Scholar

[2]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405.  Google Scholar

[3]

H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2010.  Google Scholar

[4]

H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

[5]

D. Brochet, D. Hilhorst and X. Chen, Finite-dimensional exponential attractor for the phase field model, Appl. Anal., 49 (1993), 197-212. doi: 10.1080/00036819108840173.  Google Scholar

[6]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford University Press, New York, 1998.  Google Scholar

[7]

R. Chill, E. Fašangová and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard and Caginalp equations with boundary conditions, Math. Nachr., 279 (2006), 1448-1462. doi: 10.1002/mana.200410431.  Google Scholar

[8]

R. Chill, On the Lojasiewicz-Simon gradient inequality, J. Funct. Anal., 201 (2003), 572-601. doi: 10.1016/S0022-1236(02)00102-7.  Google Scholar

[9]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[10]

E. Feireisl, F. Issard-Roch and H. Petzeltová, A non-smooth version of the Łojasiewicz- 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

[11]

E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Differential Equations, 12 (2000), 647-673. doi: 10.1023/A:1026467729263.  Google Scholar

[12]

A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations, 9 (1999), 95-124. doi: 10.1007/s005260050133.  Google Scholar

[13]

A. Haraux and M. A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. Differential Equations, 144 (1998), 302-312. doi: 10.1006/jdeq.1997.3392.  Google Scholar

[14]

A. Haraux and M. A. Jendoubi, On the convergence of global and bounded solutions of some evolution equations, J. Evol. Equ., 7 (2007), 449-470. doi: 10.1007/s00028-007-0297-8.  Google Scholar

[15]

A. Haraux, M. A. Jendoubi and O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations, J. Evol. Equ., 3 (2003), 463-484. doi: 10.1007/s00028-003-1112-8.  Google Scholar

[16]

J. K. Hale, Ordinary Differential Equations, First edition, Wiley, New York, 1969.  Google Scholar

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981.  Google Scholar

[18]

S. Huang, Gradient Inequalities. With Application to Asymptotic Behavior and Stability of Gradient-Like, Mathematical Surveys and Monographs, 126, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/126.  Google Scholar

[19]

S. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal. Ser. A: Theory Methods, 46 (2001), 675-698. doi: 10.1016/S0362-546X(00)00145-0.  Google Scholar

[20]

M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153 (1998), 187-202. doi: 10.1006/jfan.1997.3174.  Google Scholar

[21]

M. A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. Differential Equations, 144 (1998), 302-312. doi: 10.1006/jdeq.1997.3392.  Google Scholar

[22]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1968. Google Scholar

[23]

S. 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

[24]

S. Łojasiewicz, Ensembles Semi-Analytiques, I.H.E.S., Bures-sur-Yvette, 1965. Google Scholar

[25]

S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques reéls, Colloques Internationaux du C.N.R.S, Les Equations aux Derivées Partielles, 117 (1963), 87-89.  Google Scholar

[26]

S. Łojasiewicz, Sur la géométrie semi- et sous-analytique, Ann. Inst. Fourier (Grenoble), 43 (1993), 1575-1595. doi: 10.5802/aif.1384.  Google Scholar

[27]

M. Marion, Attractors for reaction-diffusion equations: Existence and estimate of their dimension, Appl. Anal., 25 (1987), 101-147. doi: 10.1080/00036818708839678.  Google Scholar

[28]

J. C. Robinson, Infinite-dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

[29]

J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. of Appl. Math., 48 (1992), 249-264. doi: 10.1093/imamat/48.3.249.  Google Scholar

[30]

P. Rybka and K.-H. Hoffmann, Convergence of solutions to Cahn-Hilliard equation, Commun. PDE., 24 (1999), 1055-1077. doi: 10.1080/03605309908821458.  Google Scholar

[31]

L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. of Math., 118 (1983), 525-571. doi: 10.2307/2006981.  Google Scholar

[32]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. H. F. Fournier, Sobolev Spaces, $2^{nd}$ edition, Academic Press, 2003.  Google Scholar

[2]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405.  Google Scholar

[3]

H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2010.  Google Scholar

[4]

H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

[5]

D. Brochet, D. Hilhorst and X. Chen, Finite-dimensional exponential attractor for the phase field model, Appl. Anal., 49 (1993), 197-212. doi: 10.1080/00036819108840173.  Google Scholar

[6]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford University Press, New York, 1998.  Google Scholar

[7]

R. Chill, E. Fašangová and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard and Caginalp equations with boundary conditions, Math. Nachr., 279 (2006), 1448-1462. doi: 10.1002/mana.200410431.  Google Scholar

[8]

R. Chill, On the Lojasiewicz-Simon gradient inequality, J. Funct. Anal., 201 (2003), 572-601. doi: 10.1016/S0022-1236(02)00102-7.  Google Scholar

[9]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[10]

E. Feireisl, F. Issard-Roch and H. Petzeltová, A non-smooth version of the Łojasiewicz- 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

[11]

E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Differential Equations, 12 (2000), 647-673. doi: 10.1023/A:1026467729263.  Google Scholar

[12]

A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations, 9 (1999), 95-124. doi: 10.1007/s005260050133.  Google Scholar

[13]

A. Haraux and M. A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. Differential Equations, 144 (1998), 302-312. doi: 10.1006/jdeq.1997.3392.  Google Scholar

[14]

A. Haraux and M. A. Jendoubi, On the convergence of global and bounded solutions of some evolution equations, J. Evol. Equ., 7 (2007), 449-470. doi: 10.1007/s00028-007-0297-8.  Google Scholar

[15]

A. Haraux, M. A. Jendoubi and O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations, J. Evol. Equ., 3 (2003), 463-484. doi: 10.1007/s00028-003-1112-8.  Google Scholar

[16]

J. K. Hale, Ordinary Differential Equations, First edition, Wiley, New York, 1969.  Google Scholar

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981.  Google Scholar

[18]

S. Huang, Gradient Inequalities. With Application to Asymptotic Behavior and Stability of Gradient-Like, Mathematical Surveys and Monographs, 126, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/126.  Google Scholar

[19]

S. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal. Ser. A: Theory Methods, 46 (2001), 675-698. doi: 10.1016/S0362-546X(00)00145-0.  Google Scholar

[20]

M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153 (1998), 187-202. doi: 10.1006/jfan.1997.3174.  Google Scholar

[21]

M. A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. Differential Equations, 144 (1998), 302-312. doi: 10.1006/jdeq.1997.3392.  Google Scholar

[22]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1968. Google Scholar

[23]

S. 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

[24]

S. Łojasiewicz, Ensembles Semi-Analytiques, I.H.E.S., Bures-sur-Yvette, 1965. Google Scholar

[25]

S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques reéls, Colloques Internationaux du C.N.R.S, Les Equations aux Derivées Partielles, 117 (1963), 87-89.  Google Scholar

[26]

S. Łojasiewicz, Sur la géométrie semi- et sous-analytique, Ann. Inst. Fourier (Grenoble), 43 (1993), 1575-1595. doi: 10.5802/aif.1384.  Google Scholar

[27]

M. Marion, Attractors for reaction-diffusion equations: Existence and estimate of their dimension, Appl. Anal., 25 (1987), 101-147. doi: 10.1080/00036818708839678.  Google Scholar

[28]

J. C. Robinson, Infinite-dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

[29]

J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. of Appl. Math., 48 (1992), 249-264. doi: 10.1093/imamat/48.3.249.  Google Scholar

[30]

P. Rybka and K.-H. Hoffmann, Convergence of solutions to Cahn-Hilliard equation, Commun. PDE., 24 (1999), 1055-1077. doi: 10.1080/03605309908821458.  Google Scholar

[31]

L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. of Math., 118 (1983), 525-571. doi: 10.2307/2006981.  Google Scholar

[32]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

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