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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, (2003).

[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. doi: 10.1002/cpa.3160120405.

[3]

H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext, (2010).

[4]

H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert,, North-Holland Publishing Co., (1973).

[5]

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

[6]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford University Press, (1998).

[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. doi: 10.1002/mana.200410431.

[8]

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

[9]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).

[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. doi: 10.1016/j.jde.2003.10.026.

[11]

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

[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. doi: 10.1007/s005260050133.

[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. doi: 10.1006/jdeq.1997.3392.

[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. doi: 10.1007/s00028-007-0297-8.

[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. doi: 10.1007/s00028-003-1112-8.

[16]

J. K. Hale, Ordinary Differential Equations,, First edition, (1969).

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer, (1981).

[18]

S. Huang, Gradient Inequalities. With Application to Asymptotic Behavior and Stability of Gradient-Like,, Mathematical Surveys and Monographs, (2006). doi: 10.1090/surv/126.

[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. doi: 10.1016/S0362-546X(00)00145-0.

[20]

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

[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. doi: 10.1006/jdeq.1997.3392.

[22]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type,, Translations of Mathematical Monographs, (1968).

[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. doi: 10.3934/dcdss.2011.4.653.

[24]

S. Łojasiewicz, Ensembles Semi-Analytiques,, I.H.E.S., (1965).

[25]

S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques reéls,, Colloques Internationaux du C.N.R.S, 117 (1963), 87.

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

E. Zeidler, Nonlinear Functional Analysis and Its Applications,, Springer-Verlag, (1990). doi: 10.1007/978-1-4612-0985-0.

show all references

References:
[1]

R. A. Adams and J. H. F. Fournier, Sobolev Spaces,, $2^{nd}$ edition, (2003).

[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. doi: 10.1002/cpa.3160120405.

[3]

H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext, (2010).

[4]

H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert,, North-Holland Publishing Co., (1973).

[5]

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

[6]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford University Press, (1998).

[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. doi: 10.1002/mana.200410431.

[8]

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

[9]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).

[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. doi: 10.1016/j.jde.2003.10.026.

[11]

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

[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. doi: 10.1007/s005260050133.

[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. doi: 10.1006/jdeq.1997.3392.

[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. doi: 10.1007/s00028-007-0297-8.

[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. doi: 10.1007/s00028-003-1112-8.

[16]

J. K. Hale, Ordinary Differential Equations,, First edition, (1969).

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer, (1981).

[18]

S. Huang, Gradient Inequalities. With Application to Asymptotic Behavior and Stability of Gradient-Like,, Mathematical Surveys and Monographs, (2006). doi: 10.1090/surv/126.

[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. doi: 10.1016/S0362-546X(00)00145-0.

[20]

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

[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. doi: 10.1006/jdeq.1997.3392.

[22]

O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type,, Translations of Mathematical Monographs, (1968).

[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. doi: 10.3934/dcdss.2011.4.653.

[24]

S. Łojasiewicz, Ensembles Semi-Analytiques,, I.H.E.S., (1965).

[25]

S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques reéls,, Colloques Internationaux du C.N.R.S, 117 (1963), 87.

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

E. Zeidler, Nonlinear Functional Analysis and Its Applications,, Springer-Verlag, (1990). doi: 10.1007/978-1-4612-0985-0.

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