A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion

Perla El Kettani; Danielle Hilhorst; Kai Lee

doi:10.3934/dcds.2018246

Article Contents

2018, Volume 38, Issue 11: 5615-5648. Doi: 10.3934/dcds.2018246

This issue Previous Article Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry Next Article Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation

A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion

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
* Corresponding author: Danielle Hilhorst

Received: November 30, 2017

Revised: May 31, 2018

Published: August 2018

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Full Text(HTML)

Related Papers

Cited by

References

	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.
	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.
	C. Bennett and R. C Sharpley, Interpolation of Operators, volume 129. Academic press, 1988.
	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.
	W. Cheney, Analysis for Applied Mathematics, Springer, 2001. doi: 10.1007/978-1-4757-3559-8.
	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.
	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge university press, 2014. doi: 10.1017/CBO9781107295513.
	T. Funaki and S. Yokoyama, Sharp interface limit for stochastically perturbed mass conserving allen-cahn equation, arXiv preprint, arXiv: 1610.01263, 2016.
	L. Gawarecki and V. Mandrekar, Stochastic Differential Equations in Infinite Dimensions, Springer, 2011. doi: 10.1007/978-3-642-16194-0.
	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.
	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.
	N. V. Krylov and B. L. Rozovskii , Stochastic evolution equations. stochastic differential equations: Theory and applications, Journal of Soviet Mathematics, 14 (1981) , 1233-1277.
	H. H. Kuo, Introduction to Stochastic Integration, Springer Science & Business Media, 2006.
	M. Marion , Attractors for reaction-diffusion equations: Existence and estimate of their dimension, Applicable Analysis, 25 (1987) , 101-147. doi: 10.1080/00036818708839678.
	C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, volume 1905. Springer, 2007.
	M. Reiß, Stochastic Differential Equations, Lecture Notes, Humboldt University Berlin, 2003.
	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.
	R. Temam, Navier-stokes Equations, volume 2. North-Holland Amsterdam, revised edition 1979.