American Institute of Mathematical Sciences

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March  2022, 12(1): 147-168. doi: 10.3934/mcrf.2021005

Noise and stability in reaction-diffusion equations

Guangying Lv 1, , Jinlong Wei 2,, and  Guang-an Zou 3,

1. 

College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2. 

School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China
3. 

Institute of Applied Mathematics, Henan University, Kaifeng 475001, China

* Corresponding author: Jinlong Wei

Received  August 2020 Revised  November 2020 Published  March 2022 Early access  March 2021

Fund Project: The authors are grateful to the referees for their valuable suggestions and comments on the original manuscript. This research was partly supported by the NSF of China grants 11771123, 11501577, 11626085 and the Startup Foundation for Introducing Talent of NUIST

We study the stability of reaction-diffusion equations in presence of noise. The relationship of stability of solutions between the stochastic ordinary different equations and the corresponding stochastic reaction-diffusion equation is firstly established. Then, by using the Lyapunov method, sufficient conditions for mean square and stochastic stability are given. The results show that the multiplicative noise can make the solution stable, but the additive noise will be not.

Keywords: Stochastic stability, mean square stability, noise, Lyapunov method.
Mathematics Subject Classification: Primary: 35B35; Secondary: 60H15.
Citation: Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control and Related Fields, 2022, 12 (1) : 147-168. doi: 10.3934/mcrf.2021005
References:
[1]

L. Arnold, I. Chueshov and G. Ochs, Random Dynamical Systems Methods in Ship Stability: A Case Study. Interacting Stochastic Systems, Springer, Berlin, 2005, 409–433. doi: 10.1007/3-540-27110-4_19.

[2]

V. Barbu, C. Lefter and G. Tessitore, A note on the stabilizability of stochastic heat equations with multiplicative noise, C. R. Math. Acad. Sci. Paris, 334 (2002), 311-316.  doi: 10.1016/S1631-073X(02)02259-8.

[3]

V. Barbu and G. Da Prato, Internal stabilization by noise of the Navier-Stokes equation, SIAM J. Control Optim., 49 (2011), 1-20.  doi: 10.1137/09077607X.

[4]

V. Barbu, Note on the internal stabilization of stochastic parabolic equations with linearly multiplicative Gaussian noise, ESAIM Control Optim. Calc. Var., 19 (2013), 1055-1063.  doi: 10.1051/cocv/2012044.

[5]

P. -L. Chow, Stochastic partial differential equations. Applied Mathematics and Nonlinear Science Series, Chapman & Hall, Boca Raton, FL, 2007.

[6]

P-L. Chow, Explosive solutions of stochastic reaction-diffusion equations in mean $L^p$-norm, J. Differential Equations, 250 (2011), 2567-2580.  doi: 10.1016/j.jde.2010.11.008.

[7]

T. Caraballo, P. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.

[8] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. 
[9]

K. Dareiotis, M. Gerencsér and B. Gess, Entropy solutions for stochastic porous media equations, J. Differential Equations, 266 (2019), 3732-3763.  doi: 10.1016/j.jde.2018.09.012.

[10]

E. Fedrizzi and F. Flandoli, Noise prevents singularities in linear transport equations, J. Funct. Anal., 264 (2013), 1329-1354.  doi: 10.1016/j.jfa.2013.01.003.

[11]

F. Flandoli, M. Gubinelli and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), 1-53.  doi: 10.1007/s00222-009-0224-4.

[12]

F. Flandoli and D. Luo, Euler-Lagrangian approach to 3D stochastic Euler equations, J. Geom. Mech., 11 (2019), 153-165.  doi: 10.3934/jgm.2019008.

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[14]

R. Khasminskii, Stochastic Stability of Differential Equations, Stochastic Modelling and Applied Probability, vol. 66, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.

[15]

K. Liu and X. Mao, Exponential stability of non-linear stochastic evolution equations, Stochastic Process. Appl., 78 (1998), 173-193.  doi: 10.1016/S0304-4149(98)00048-9.

[16]

G. Lv and J. Duan, Impacts of noise on a class of partial differential equations, J. Differential Equations, 258 (2015), 2196-2220.  doi: 10.1016/j.jde.2014.12.002.

[17]

G. Lv, J. Duan, L. Wang and J. Wu, Impact of noise on ordinary differential equations, Dynamic system and Applications, 27 (2018), 225-236. 

[18]

G. Lv, H. Gao, J. Wei and J.-L. Wu, BMO and Morrey-Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations, J. Differential Equations, 266 (2019), 2666-2717.  doi: 10.1016/j.jde.2018.08.042.

[19]

G. Lv and J. Wei, Global existence and non-existence of stochastic parabolic equations, preprint, 2019, arXiv: 1902.07389.

[20]

M. Mackey and G. Nechaeva, Noise and stability in differential delay equations, J. Dynam. Differential Equations, 6 (1994), 395-426.  doi: 10.1007/BF02218856.

[21]

X. Mao, Exponential Stability of Stochastic Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 182. Marcel Dekker, Inc., New York, 1994.

[22]

D. H. Nguen and G. Yin, Stability of regime-switching diffusion systems with discrete states belonging to a countable set, SIAM J. Control Optim., 56 (2018), 3893-3917.  doi: 10.1137/17M1118476.

[23]

R. Tian, J. Wei and J. Wu, On a generalized population dynamics equation with environmental noise, Statist. Probab. Lett., 168 (2021) 108944, 7 pp. doi: 10.1016/j. spl. 2020.108944.

[24]

J. B. Walsh, An Introduction to Stochastic Partial Differential Equations, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986. doi: 10.1007/BFb0074920.

[25]

Y. Wang, F. Wu and X. Mao, Stability in distribution of stochastic functional differential equations, Systems Control Lett., 132 (2019), 104513, 10 pp. doi: 10.1016/j. sysconle. 2019.104513.

[26]

F. Wu, G. Yin and L.Y. Wang, Stability of a pure random delay system with two-time-scale Markovian switching, J. Differential Equations, 253 (2012), 878-905.  doi: 10.1016/j.jde.2012.04.017.

[27] Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations, Science Press, Beijing, 1990. 
[28]

X. Zhang, Stochastic differential equations with Sobolev drifts and driven by $\alpha$-stable processes, Ann. Inst. Henri Poincaré Probab. Stat., 49 (2013), 1057-1079.  doi: 10.1214/12-AIHP476.

show all references

References:
[1]

L. Arnold, I. Chueshov and G. Ochs, Random Dynamical Systems Methods in Ship Stability: A Case Study. Interacting Stochastic Systems, Springer, Berlin, 2005, 409–433. doi: 10.1007/3-540-27110-4_19.

[2]

V. Barbu, C. Lefter and G. Tessitore, A note on the stabilizability of stochastic heat equations with multiplicative noise, C. R. Math. Acad. Sci. Paris, 334 (2002), 311-316.  doi: 10.1016/S1631-073X(02)02259-8.

[3]

V. Barbu and G. Da Prato, Internal stabilization by noise of the Navier-Stokes equation, SIAM J. Control Optim., 49 (2011), 1-20.  doi: 10.1137/09077607X.

[4]

V. Barbu, Note on the internal stabilization of stochastic parabolic equations with linearly multiplicative Gaussian noise, ESAIM Control Optim. Calc. Var., 19 (2013), 1055-1063.  doi: 10.1051/cocv/2012044.

[5]

P. -L. Chow, Stochastic partial differential equations. Applied Mathematics and Nonlinear Science Series, Chapman & Hall, Boca Raton, FL, 2007.

[6]

P-L. Chow, Explosive solutions of stochastic reaction-diffusion equations in mean $L^p$-norm, J. Differential Equations, 250 (2011), 2567-2580.  doi: 10.1016/j.jde.2010.11.008.

[7]

T. Caraballo, P. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.

[8] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. 
[9]

K. Dareiotis, M. Gerencsér and B. Gess, Entropy solutions for stochastic porous media equations, J. Differential Equations, 266 (2019), 3732-3763.  doi: 10.1016/j.jde.2018.09.012.

[10]

E. Fedrizzi and F. Flandoli, Noise prevents singularities in linear transport equations, J. Funct. Anal., 264 (2013), 1329-1354.  doi: 10.1016/j.jfa.2013.01.003.

[11]

F. Flandoli, M. Gubinelli and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), 1-53.  doi: 10.1007/s00222-009-0224-4.

[12]

F. Flandoli and D. Luo, Euler-Lagrangian approach to 3D stochastic Euler equations, J. Geom. Mech., 11 (2019), 153-165.  doi: 10.3934/jgm.2019008.

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[14]

R. Khasminskii, Stochastic Stability of Differential Equations, Stochastic Modelling and Applied Probability, vol. 66, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.

[15]

K. Liu and X. Mao, Exponential stability of non-linear stochastic evolution equations, Stochastic Process. Appl., 78 (1998), 173-193.  doi: 10.1016/S0304-4149(98)00048-9.

[16]

G. Lv and J. Duan, Impacts of noise on a class of partial differential equations, J. Differential Equations, 258 (2015), 2196-2220.  doi: 10.1016/j.jde.2014.12.002.

[17]

G. Lv, J. Duan, L. Wang and J. Wu, Impact of noise on ordinary differential equations, Dynamic system and Applications, 27 (2018), 225-236. 

[18]

G. Lv, H. Gao, J. Wei and J.-L. Wu, BMO and Morrey-Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations, J. Differential Equations, 266 (2019), 2666-2717.  doi: 10.1016/j.jde.2018.08.042.

[19]

G. Lv and J. Wei, Global existence and non-existence of stochastic parabolic equations, preprint, 2019, arXiv: 1902.07389.

[20]

M. Mackey and G. Nechaeva, Noise and stability in differential delay equations, J. Dynam. Differential Equations, 6 (1994), 395-426.  doi: 10.1007/BF02218856.

[21]

X. Mao, Exponential Stability of Stochastic Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 182. Marcel Dekker, Inc., New York, 1994.

[22]

D. H. Nguen and G. Yin, Stability of regime-switching diffusion systems with discrete states belonging to a countable set, SIAM J. Control Optim., 56 (2018), 3893-3917.  doi: 10.1137/17M1118476.

[23]

R. Tian, J. Wei and J. Wu, On a generalized population dynamics equation with environmental noise, Statist. Probab. Lett., 168 (2021) 108944, 7 pp. doi: 10.1016/j. spl. 2020.108944.

[24]

J. B. Walsh, An Introduction to Stochastic Partial Differential Equations, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986. doi: 10.1007/BFb0074920.

[25]

Y. Wang, F. Wu and X. Mao, Stability in distribution of stochastic functional differential equations, Systems Control Lett., 132 (2019), 104513, 10 pp. doi: 10.1016/j. sysconle. 2019.104513.

[26]

F. Wu, G. Yin and L.Y. Wang, Stability of a pure random delay system with two-time-scale Markovian switching, J. Differential Equations, 253 (2012), 878-905.  doi: 10.1016/j.jde.2012.04.017.

[27] Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations, Science Press, Beijing, 1990. 
[28]

X. Zhang, Stochastic differential equations with Sobolev drifts and driven by $\alpha$-stable processes, Ann. Inst. Henri Poincaré Probab. Stat., 49 (2013), 1057-1079.  doi: 10.1214/12-AIHP476.

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