2012, 1(1): 43-56. doi: 10.3934/eect.2012.1.43

Invariance for stochastic reaction-diffusion equations

1. 

Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scienti ca 1, I-00133 Roma, Italy

2. 

Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, I-56125 Pisa, Italy

Received  December 2011 Revised  February 2012 Published  March 2012

Given a stochastic reaction-diffusion equation on a bounded open subset $\mathcal O$ of $\mathbb{R}^n$, we discuss conditions for the invariance of a nonempty closed convex subset $K$ of $L^2(\mathcal O)$ under the corresponding flow. We obtain two general results under the assumption that the fourth power of the distance from $K$ is of class $C^2$, providing, respectively, a necessary and a sufficient condition for invariance. We also study the example where $K$ is the cone of all nonnegative functions in $L^2(\mathcal O)$.
Citation: Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43
References:
[1]

S. Agmon, "Lectures on Elliptic Boundary Value Problems,", Prepared for publication by B. Frank Jones, (1965).

[2]

P. Cannarsa and G. Da Prato, Stochastic viability for regular closed sets in Hilbert spaces,, Rend. Lincei Math. Appl., 22 (2011), 1.

[3]

S. Cerrai, Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term,, Probab. Theory Related Fields, 125 (2003), 271. doi: 10.1007/s00440-002-0230-6.

[4]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Encyclopedia of Mathematics and its Applications, 44 (1992).

[5]

G. Da Prato and J. Zabczyk, "Second Order Partial Differential Equations in Hilbert Spaces,", London Mathematical Society Lecture Notes, 293 (2002).

[6]

L. Evans and R. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992).

[7]

D. Grieser, Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary,, Comm. Partial Differential Equations, 27 (2002), 1283.

[8]

S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert spaces,, Ann. Probab., 23 (1995), 157. doi: 10.1214/aop/1176988381.

show all references

References:
[1]

S. Agmon, "Lectures on Elliptic Boundary Value Problems,", Prepared for publication by B. Frank Jones, (1965).

[2]

P. Cannarsa and G. Da Prato, Stochastic viability for regular closed sets in Hilbert spaces,, Rend. Lincei Math. Appl., 22 (2011), 1.

[3]

S. Cerrai, Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term,, Probab. Theory Related Fields, 125 (2003), 271. doi: 10.1007/s00440-002-0230-6.

[4]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Encyclopedia of Mathematics and its Applications, 44 (1992).

[5]

G. Da Prato and J. Zabczyk, "Second Order Partial Differential Equations in Hilbert Spaces,", London Mathematical Society Lecture Notes, 293 (2002).

[6]

L. Evans and R. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992).

[7]

D. Grieser, Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary,, Comm. Partial Differential Equations, 27 (2002), 1283.

[8]

S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert spaces,, Ann. Probab., 23 (1995), 157. doi: 10.1214/aop/1176988381.

[1]

Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281

[2]

Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 187-208. doi: 10.3934/dcds.2018009

[3]

Igor Chueshov, Michael Scheutzow. Invariance and monotonicity for stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1533-1554. doi: 10.3934/dcdsb.2013.18.1533

[4]

Wei Wang, Anthony Roberts. Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 253-273. doi: 10.3934/dcds.2011.31.253

[5]

Ricardo Enguiça, Andrea Gavioli, Luís Sanchez. A class of singular first order differential equations with applications in reaction-diffusion. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 173-191. doi: 10.3934/dcds.2013.33.173

[6]

Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351

[7]

Ciprian G. Gal, Mahamadi Warma. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1279-1319. doi: 10.3934/dcds.2016.36.1279

[8]

Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 19-61. doi: 10.3934/dcds.2009.25.19

[9]

Fuzhi Li, Yangrong Li, Renhai Wang. Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3663-3685. doi: 10.3934/dcds.2018158

[10]

Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 41-67. doi: 10.3934/dcds.2008.21.41

[11]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

[12]

Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3629-3651. doi: 10.3934/dcdsb.2017143

[13]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229

[14]

Peter E. Kloeden, Thomas Lorenz, Meihua Yang. Reaction-diffusion equations with a switched--off reaction zone. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1907-1933. doi: 10.3934/cpaa.2014.13.1907

[15]

María Anguiano, Tomás Caraballo, José Real, José Valero. Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 307-326. doi: 10.3934/dcdsb.2010.14.307

[16]

Jong-Shenq Guo, Yoshihisa Morita. Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 193-212. doi: 10.3934/dcds.2005.12.193

[17]

Jacson Simsen, Mariza Stefanello Simsen, Marcos Roberto Teixeira Primo. Reaction-Diffusion equations with spatially variable exponents and large diffusion. Communications on Pure & Applied Analysis, 2016, 15 (2) : 495-506. doi: 10.3934/cpaa.2016.15.495

[18]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295

[19]

Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203

[20]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221

2016 Impact Factor: 0.826

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]