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November  2018, 38(11): 5615-5648. doi: 10.3934/dcds.2018246

A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion

Perla El Kettani 1, , Danielle Hilhorst 2,, and  Kai Lee 3,

1. 

Laboratoire de Mathématique, Analyse Numérique et EDP, University of Paris-Sud, F-91405 Orsay Cedex, France
2. 

CNRS and Laboratoire de Mathématique, Analyse Numérique et EDP, University of Paris-Sud, F-91405 Orsay Cedex, France
3. 

Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan

* Corresponding author: Danielle Hilhorst

Received  December 2017 Revised  June 2018 Published  August 2018

Fund Project: The first author is supported by a public grant as part of the Investissement d'avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH.

In this paper, we prove a well posedness result for an initial boundary value problem for a stochastic nonlocal reaction-diffusion equation with nonlinear diffusion together with a nul-flux boundary condition in an open bounded domain of $\mathbb{R}^n$ with a smooth boundary. We suppose that the additive noise is induced by a Q-Brownian motion.

Keywords: Stochastic nonlocal reaction-diffusion equation, monotonicity method, conservation of mass.
Mathematics Subject Classification: Primary: 60H15, 60H30, 35K55, 35K57.
Citation: Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246
References:
[1]

D. C. Antonopoulou, P. W. Bates, D. Blömker and G. D. Karali, Motion of a droplet for the stochastic mass-conserving allen-cahn equation, SIAM Journal on Mathematical Analysis, 48 (2016), 670-708.  doi: 10.1137/151005105.  Google Scholar

[2]

C. Bauzet, G. Vallet and P. Wittbold, The cauchy problem for conservation laws with a multiplicative stochastic perturbation, Journal of Hyperbolic Differential Equations, 9 (2012), 661-709.  doi: 10.1142/S0219891612500221.  Google Scholar

[3]

C. Bennett and R. C Sharpley, Interpolation of Operators, volume 129. Academic press, 1988.  Google Scholar

[4]

S. Boussaïd, D. Hilhorst and T. N. Nguyen, Convergence to steady states for solutions of a reaction-diffusion equation, Evol. Equ. Control Theory, 4 (2015), 39-59.  doi: 10.3934/eect.2015.4.39.  Google Scholar

[5]

W. Cheney, Analysis for Applied Mathematics, Springer, 2001. doi: 10.1007/978-1-4757-3559-8.  Google Scholar

[6]

G. Da Prato and A. Debussche, Stochastic cahn-hilliard equation, Nonlinear Analysis: Theory, Methods & Applications, 26 (1996), 241-263.  doi: 10.1016/0362-546X(94)00277-O.  Google Scholar

[7]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge university press, 2014. doi: 10.1017/CBO9781107295513.  Google Scholar

[8]

T. Funaki and S. Yokoyama, Sharp interface limit for stochastically perturbed mass conserving allen-cahn equation, arXiv preprint, arXiv: 1610.01263, 2016. Google Scholar

[9]

L. Gawarecki and V. Mandrekar, Stochastic Differential Equations in Infinite Dimensions, Springer, 2011. doi: 10.1007/978-3-642-16194-0.  Google Scholar

[10]

B. Gess, Strong solutions for stochastic partial differential equations of gradient type, Journal of Functional Analysis, 263 (2012), 2355-2383.  doi: 10.1016/j.jfa.2012.07.001.  Google Scholar

[11]

I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, volume 113. Springer Science & Business Media, 2013. doi: 10.1007/978-3-642-31898-6.  Google Scholar

[12]

N. V. Krylov and B. L. Rozovskii, Stochastic evolution equations. stochastic differential equations: Theory and applications, Journal of Soviet Mathematics, 14 (1981), 1233-1277.   Google Scholar

[13]

H. H. Kuo, Introduction to Stochastic Integration, Springer Science & Business Media, 2006.  Google Scholar

[14]

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

[15]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, volume 1905. Springer, 2007.  Google Scholar

[16]

M. Reiß, Stochastic Differential Equations, Lecture Notes, Humboldt University Berlin, 2003. Google Scholar

[17]

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

[18]

R. Temam, Navier-stokes Equations, volume 2. North-Holland Amsterdam, revised edition 1979.  Google Scholar

show all references

References:
[1]

D. C. Antonopoulou, P. W. Bates, D. Blömker and G. D. Karali, Motion of a droplet for the stochastic mass-conserving allen-cahn equation, SIAM Journal on Mathematical Analysis, 48 (2016), 670-708.  doi: 10.1137/151005105.  Google Scholar

[2]

C. Bauzet, G. Vallet and P. Wittbold, The cauchy problem for conservation laws with a multiplicative stochastic perturbation, Journal of Hyperbolic Differential Equations, 9 (2012), 661-709.  doi: 10.1142/S0219891612500221.  Google Scholar

[3]

C. Bennett and R. C Sharpley, Interpolation of Operators, volume 129. Academic press, 1988.  Google Scholar

[4]

S. Boussaïd, D. Hilhorst and T. N. Nguyen, Convergence to steady states for solutions of a reaction-diffusion equation, Evol. Equ. Control Theory, 4 (2015), 39-59.  doi: 10.3934/eect.2015.4.39.  Google Scholar

[5]

W. Cheney, Analysis for Applied Mathematics, Springer, 2001. doi: 10.1007/978-1-4757-3559-8.  Google Scholar

[6]

G. Da Prato and A. Debussche, Stochastic cahn-hilliard equation, Nonlinear Analysis: Theory, Methods & Applications, 26 (1996), 241-263.  doi: 10.1016/0362-546X(94)00277-O.  Google Scholar

[7]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge university press, 2014. doi: 10.1017/CBO9781107295513.  Google Scholar

[8]

T. Funaki and S. Yokoyama, Sharp interface limit for stochastically perturbed mass conserving allen-cahn equation, arXiv preprint, arXiv: 1610.01263, 2016. Google Scholar

[9]

L. Gawarecki and V. Mandrekar, Stochastic Differential Equations in Infinite Dimensions, Springer, 2011. doi: 10.1007/978-3-642-16194-0.  Google Scholar

[10]

B. Gess, Strong solutions for stochastic partial differential equations of gradient type, Journal of Functional Analysis, 263 (2012), 2355-2383.  doi: 10.1016/j.jfa.2012.07.001.  Google Scholar

[11]

I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, volume 113. Springer Science & Business Media, 2013. doi: 10.1007/978-3-642-31898-6.  Google Scholar

[12]

N. V. Krylov and B. L. Rozovskii, Stochastic evolution equations. stochastic differential equations: Theory and applications, Journal of Soviet Mathematics, 14 (1981), 1233-1277.   Google Scholar

[13]

H. H. Kuo, Introduction to Stochastic Integration, Springer Science & Business Media, 2006.  Google Scholar

[14]

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

[15]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, volume 1905. Springer, 2007.  Google Scholar

[16]

M. Reiß, Stochastic Differential Equations, Lecture Notes, Humboldt University Berlin, 2003. Google Scholar

[17]

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

[18]

R. Temam, Navier-stokes Equations, volume 2. North-Holland Amsterdam, revised edition 1979.  Google Scholar

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