August  2019, 24(8): 4055-4078. doi: 10.3934/dcdsb.2019050

Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions

1. 

Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria

2. 

Department of Mathematics, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands

3. 

Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria

4. 

School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Dai Co Viet No. 1, Hanoi, Vietnam

* Corresponding author: Stefanie Sonner

Received  March 2018 Revised  August 2018 Published  February 2019

Fund Project: Our research was supported by the ASEAN-European Academic University Network (ASEA-UNINET), and partially supported by NAWI Graz, IGDK 1754, and NAFOSTED project 101.01-2017.302

The stabilisation by noise on the boundary of the Chafee-Infante equation with dynamical boundary conditions subject to a multiplicative Itô noise is studied. In particular, we show that there exists a finite range of noise intensities that imply the exponential stability of the trivial steady state. This differs from previous works on the stabilisation by noise of parabolic PDEs, where the noise acts inside the domain and stabilisation typically occurs for an infinite range of noise intensities. To the best of our knowledge, this is the first result on the stabilisation of PDEs by boundary noise.

Citation: Klemens Fellner, Stefanie Sonner, Bao Quoc Tang, Do Duc Thuan. Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4055-4078. doi: 10.3934/dcdsb.2019050
References:
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R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition, Elsevier/Academic Press, Amsterdam, 2003. Google Scholar

[2]

E. Alòs and S. Bonaccorsi, Stochastic partial differential equations with Dirichlet white-noise boundary conditions, Ann. Inst. H. Poincaré Probab. Statist., 38 (2002), 125-154. doi: 10.1016/S0246-0203(01)01097-4. Google Scholar

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E. Alòs and S. Bonaccorsi, Stability for stochastic partial differential equations with Dirichlet white-noise boundary conditions, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5 (2002), 465-481. doi: 10.1142/S0219025702000948. Google Scholar

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T. Caraballo, Recent results on stabilization of PDEs with noise, Bol. Soc. Esp. Mat. Apl., 37 (2006), 47-70. Google Scholar

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T. CaraballoH. CrauelJ. Langa and J. Robinson, The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proceedings of the American Mathematical Society, 135 (2007), 373-382. doi: 10.1090/S0002-9939-06-08593-5. Google Scholar

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T. Caraballo and P. E. Kloeden, Stabilization of evolution equations by noise, Interdiscip. Math. Sci., 8 (2010), World Sci. Publ., Hackensack, NJ, 43–66. doi: 10.1142/9789814277266_0003. Google Scholar

[8]

T. CaraballoP. E. Kloeden and B. Schmalfuß, Stabilization of stationary solutions of evolution equations by noise, Discrete Conts. Dyn. Systems, Series B., 6 (2006), 1199-1212. doi: 10.3934/dcdsb.2006.6.1199. Google Scholar

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T. CaraballoK. Liu and X. R. Mao, On stabilization of partial differential equations by noise, Nagoya Math. J., 161 (2001), 155-170. doi: 10.1017/S0027763000022169. Google Scholar

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H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Reprint of the second edition, Oxford University Press, New York, 1988. Google Scholar

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R. Czaja and P. Marín-Rubio, Pullback exponential attractors for parabolic equations with dynamical boundary conditions, Taiwanese J. Math., 21 (2017), 819-839. doi: 10.11650/tjm/7862. Google Scholar

[12]

A. DebusscheM. Fuhrman and G. Tessitore, Optimal control of a stochastic heat equation with boundary-noise and boundary-control, ESAIM: Control, Optimisation and Calculus of Variations, 13 (2007), 178-205. doi: 10.1051/cocv:2007001. Google Scholar

[13]

J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364. doi: 10.1080/03605309308820976. Google Scholar

[14]

A. Filinovskiy, On the eigenvalues of a Robin problem with a large parameter, Mathematica Bohemica, 139 (2014), 341-352. Google Scholar

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A. Filinovskiy, On the Asymptotic Behavior of the First Eigenvalue of Robin Problem With Large Parameter, J. Elliptic Para. Equations, 1 (2015), 123-135. doi: 10.1007/BF03377372. Google Scholar

[16]

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

[17]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. Google Scholar

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Lecture Notes in Mathematics, 1981. Google Scholar

[19]

H. Kovařík, On the Lowest Eigenvalue of Laplace Operators with Mixed Boundary Conditions, J. Geom. Anal., 24 (2014), 1509-1525. doi: 10.1007/s12220-012-9383-4. Google Scholar

[20]

A. A. Kwiecinska, Stabilization of partial differential equations by noise, Stoch. Proc. & Appl., 79 (1999), 179-184. doi: 10.1016/S0304-4149(98)00080-5. Google Scholar

[21]

J.-L. Lions, Quelques Méthodes de Résolution des Probl'emes aux Limites non Linéaires, (French), Dunod; Gauthier-Villars, Paris, 1969. Google Scholar

[22]

K. Liu, On stability for a class of semilinear stochastic evolution equations, Stochastic Process. Appl., 70 (1997), 219-241. doi: 10.1016/S0304-4149(97)00062-8. Google Scholar

[23]

K. Lu, A Hartman-Grobman theorem for scalar reaction-diffusion equations, J. Diff. Eqs., 93 (1991), 364-394. doi: 10.1016/0022-0396(91)90017-4. Google Scholar

[24]

X. R. Mao, Stochastic stabilization and destabilization, Control Letters, 23 (1994), 279-290. doi: 10.1016/0167-6911(94)90050-7. Google Scholar

[25]

I. Munteanu, Stabilization of stochastic parabolic equations with boundary-noise and boundary-control, J. Math. Anal. Appl., 449 (2017), 829-842. doi: 10.1016/j.jmaa.2016.12.047. Google Scholar

[26]

E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastic Process. Appl., 3 (1979), 127-167. doi: 10.1080/17442507908833142. Google Scholar

[27]

M. Sofonea and A. Matei, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, Vol. 18, Springer Science & Business Media, 2009. Google Scholar

[28]

R. B. Sowers, Multidimensional reaction-diffusion equations with white noise boundary perturbations, Ann. Probab., 22 (1994), 2071-2121. doi: 10.1214/aop/1176988495. Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition, Elsevier/Academic Press, Amsterdam, 2003. Google Scholar

[2]

E. Alòs and S. Bonaccorsi, Stochastic partial differential equations with Dirichlet white-noise boundary conditions, Ann. Inst. H. Poincaré Probab. Statist., 38 (2002), 125-154. doi: 10.1016/S0246-0203(01)01097-4. Google Scholar

[3]

E. Alòs and S. Bonaccorsi, Stability for stochastic partial differential equations with Dirichlet white-noise boundary conditions, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5 (2002), 465-481. doi: 10.1142/S0219025702000948. Google Scholar

[4]

V. Barbu, Stabilization of Navier–Stokes Flows, Springer, London, 2011. doi: 10.1007/978-0-85729-043-4. Google Scholar

[5]

T. Caraballo, Recent results on stabilization of PDEs with noise, Bol. Soc. Esp. Mat. Apl., 37 (2006), 47-70. Google Scholar

[6]

T. CaraballoH. CrauelJ. Langa and J. Robinson, The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proceedings of the American Mathematical Society, 135 (2007), 373-382. doi: 10.1090/S0002-9939-06-08593-5. Google Scholar

[7]

T. Caraballo and P. E. Kloeden, Stabilization of evolution equations by noise, Interdiscip. Math. Sci., 8 (2010), World Sci. Publ., Hackensack, NJ, 43–66. doi: 10.1142/9789814277266_0003. Google Scholar

[8]

T. CaraballoP. E. Kloeden and B. Schmalfuß, Stabilization of stationary solutions of evolution equations by noise, Discrete Conts. Dyn. Systems, Series B., 6 (2006), 1199-1212. doi: 10.3934/dcdsb.2006.6.1199. Google Scholar

[9]

T. CaraballoK. Liu and X. R. Mao, On stabilization of partial differential equations by noise, Nagoya Math. J., 161 (2001), 155-170. doi: 10.1017/S0027763000022169. Google Scholar

[10]

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Reprint of the second edition, Oxford University Press, New York, 1988. Google Scholar

[11]

R. Czaja and P. Marín-Rubio, Pullback exponential attractors for parabolic equations with dynamical boundary conditions, Taiwanese J. Math., 21 (2017), 819-839. doi: 10.11650/tjm/7862. Google Scholar

[12]

A. DebusscheM. Fuhrman and G. Tessitore, Optimal control of a stochastic heat equation with boundary-noise and boundary-control, ESAIM: Control, Optimisation and Calculus of Variations, 13 (2007), 178-205. doi: 10.1051/cocv:2007001. Google Scholar

[13]

J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364. doi: 10.1080/03605309308820976. Google Scholar

[14]

A. Filinovskiy, On the eigenvalues of a Robin problem with a large parameter, Mathematica Bohemica, 139 (2014), 341-352. Google Scholar

[15]

A. Filinovskiy, On the Asymptotic Behavior of the First Eigenvalue of Robin Problem With Large Parameter, J. Elliptic Para. Equations, 1 (2015), 123-135. doi: 10.1007/BF03377372. Google Scholar

[16]

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

[17]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. Google Scholar

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Lecture Notes in Mathematics, 1981. Google Scholar

[19]

H. Kovařík, On the Lowest Eigenvalue of Laplace Operators with Mixed Boundary Conditions, J. Geom. Anal., 24 (2014), 1509-1525. doi: 10.1007/s12220-012-9383-4. Google Scholar

[20]

A. A. Kwiecinska, Stabilization of partial differential equations by noise, Stoch. Proc. & Appl., 79 (1999), 179-184. doi: 10.1016/S0304-4149(98)00080-5. Google Scholar

[21]

J.-L. Lions, Quelques Méthodes de Résolution des Probl'emes aux Limites non Linéaires, (French), Dunod; Gauthier-Villars, Paris, 1969. Google Scholar

[22]

K. Liu, On stability for a class of semilinear stochastic evolution equations, Stochastic Process. Appl., 70 (1997), 219-241. doi: 10.1016/S0304-4149(97)00062-8. Google Scholar

[23]

K. Lu, A Hartman-Grobman theorem for scalar reaction-diffusion equations, J. Diff. Eqs., 93 (1991), 364-394. doi: 10.1016/0022-0396(91)90017-4. Google Scholar

[24]

X. R. Mao, Stochastic stabilization and destabilization, Control Letters, 23 (1994), 279-290. doi: 10.1016/0167-6911(94)90050-7. Google Scholar

[25]

I. Munteanu, Stabilization of stochastic parabolic equations with boundary-noise and boundary-control, J. Math. Anal. Appl., 449 (2017), 829-842. doi: 10.1016/j.jmaa.2016.12.047. Google Scholar

[26]

E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastic Process. Appl., 3 (1979), 127-167. doi: 10.1080/17442507908833142. Google Scholar

[27]

M. Sofonea and A. Matei, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, Vol. 18, Springer Science & Business Media, 2009. Google Scholar

[28]

R. B. Sowers, Multidimensional reaction-diffusion equations with white noise boundary perturbations, Ann. Probab., 22 (1994), 2071-2121. doi: 10.1214/aop/1176988495. Google Scholar

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