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Stabilization of a class of semilinear degenerate parabolic equations by Itô noise. (English) Zbl 1347.60075

Summary: We investigate the effect of Itô noise on the stability of stationary solutions to a class of semilinear degenerate parabolic equations with the nonlinearity satisfying an arbitrary polynomial growth condition. We will show that an Itô noise of sufficient intensity will stabilize the unstable stationary solution.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
35R60 PDEs with randomness, stochastic partial differential equations
35B35 Stability in context of PDEs
35K65 Degenerate parabolic equations
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[1] Anh C. T., Bao T. Q. and Thanh N. V., Regularity of random attractors for stochastic semilinear degenerate parabolic equations, Electron. J. Differential Equations 2012 (2012), Paper No. 207.; Anh, C. T.; Bao, T. Q.; Thanh, N. V., Regularity of random attractors for stochastic semilinear degenerate parabolic equations, Electron. J. Differential Equations, 2012 (2012) · Zbl 1290.35354
[2] Anh C. T., Binh N. D. and Thuy L. T., On the global attractors for a class of semilinear degenerate parabolic equations, Ann. Polon. Math. 98 (2010), 71-89.; Anh, C. T.; Binh, N. D.; Thuy, L. T., On the global attractors for a class of semilinear degenerate parabolic equations, Ann. Polon. Math., 98, 71-89 (2010) · Zbl 1194.35073
[3] Arnold L., Crauel H. and Wihstutz V., Stabilization of linear systems by noise, SIAM J. Control Optim. 21 (1983), 451-461.; Arnold, L.; Crauel, H.; Wihstutz, V., Stabilization of linear systems by noise, SIAM J. Control Optim., 21, 451-461 (1983) · Zbl 0514.93069
[4] Caldiroli P. and Musina R., On a variational degenerate elliptic problem, Nonlinear Diff. Equ. Appl. 7 (2000), 187-199.; Caldiroli, P.; Musina, R., On a variational degenerate elliptic problem, Nonlinear Diff. Equ. Appl., 7, 187-199 (2000) · Zbl 0960.35039
[5] Caraballo T., Crauel H., Langa J. A. and Robinson J. C., The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proc. Amer. Math. Soc. 135 (2007), 373-382.; Caraballo, T.; Crauel, H.; Langa, J. A.; Robinson, J. C., The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proc. Amer. Math. Soc., 135, 373-382 (2007) · Zbl 1173.60022
[6] Caraballo T. and Kloeden P. E., Stabilization of evolution equations by noise, Recent Development in Stochastic Dynamics and Stochastic Analysis, Interdiscip. Math. Sci. 8, World Scientific, Hackensack (2010), 43-66.; Caraballo, T.; Kloeden, P. E., Stabilization of evolution equations by noise, Recent Development in Stochastic Dynamics and Stochastic Analysis, 43-66 (2010) · Zbl 1230.35148
[7] Caraballo T., Langa J. A. and Robinson J. C., Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Discrete Contin. Dyn. Syst. 6 (2000), 875-892.; Caraballo, T.; Langa, J. A.; Robinson, J. C., Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Discrete Contin. Dyn. Syst., 6, 875-892 (2000) · Zbl 1011.37031
[8] Caraballo T., Liu K. and Mao X., On stabilization of partial differential equations by noise, Nagoya Math. J. 161 (2001), 155-170.; Caraballo, T.; Liu, K.; Mao, X., On stabilization of partial differential equations by noise, Nagoya Math. J., 161, 155-170 (2001) · Zbl 0986.60058
[9] Caraballo T. and Robinson J. C., Stabilisation of linear PDEs by Stratonovich noise, Systems Control Lett. 53 (2004), 41-50.; Caraballo, T.; Robinson, J. C., Stabilisation of linear PDEs by Stratonovich noise, Systems Control Lett., 53, 41-50 (2004) · Zbl 1157.60332
[10] Cerrai S., Stabilization by noise for a class of stochastic reaction-diffusion equations, Probab. Theory Related Fields 133 (2005), 190-214.; Cerrai, S., Stabilization by noise for a class of stochastic reaction-diffusion equations, Probab. Theory Related Fields, 133, 190-214 (2005) · Zbl 1077.60046
[11] Da Prato G. and Zabczyk J., Stochastic Equations in Infinite Dimensions, 2nd ed., Encyclopedia Math. Appl. 152, Cambridge University Press, Cambridge, 2014.; Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions (2014) · Zbl 1317.60077
[12] Dautray R. and Lions J. L., Mathematical Analysis and Numerical Methods for Science and Technology. Vol. I: Physical Origins and Classical Methods, Springer, Berlin, 1985.; Dautray, R.; Lions, J. L., Mathematical Analysis and Numerical Methods for Science and Technology. Vol. I: Physical Origins and Classical Methods (1985) · Zbl 0942.35001
[13] Gawarecki L. and Mandrekar V., Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations, Probab. Appl. (N. Y.), Springer, Heidelberg, 2011.; Gawarecki, L.; Mandrekar, V., Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations (2011) · Zbl 1228.60002
[14] Has’minskii R. Z., Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, Netherlands, 1980.; Has’minskii, R. Z., Stochastic Stability of Differential Equations (1980)
[15] Karachalios N. I. and Zographopoulos N. B., On the dynamics of a degenerate parabolic equation: Global bifurcation of stationary states and convergence, Calc. Var. Partial Differential Equations 25 (2006), 361-393.; Karachalios, N. I.; Zographopoulos, N. B., On the dynamics of a degenerate parabolic equation: Global bifurcation of stationary states and convergence, Calc. Var. Partial Differential Equations, 25, 361-393 (2006) · Zbl 1090.35035
[16] Kwiecińska A., Stabilization of partial differential equations by noise, Stochastic Process. Appl. 79 (1999), 179-184.; Kwiecińska, A., Stabilization of partial differential equations by noise, Stochastic Process. Appl., 79, 179-184 (1999) · Zbl 0962.60052
[17] Mandrekar V. and Rüdiger B., Stochastic Integration in Banach Spaces. Theory and Applications, Probab. Theory Stoch. Model. 73, Springer, Cham, 2015.; Mandrekar, V.; Rüdiger, B., Stochastic Integration in Banach Spaces. Theory and Applications (2015) · Zbl 1314.60007
[18] Mao X. R., Stochastic stabilization and destabilization, Systems Control Lett. 23 (1994), 279-290.; Mao, X. R., Stochastic stabilization and destabilization, Systems Control Lett., 23, 279-290 (1994) · Zbl 0820.93071
[19] Lions J. L., Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, 1969.; Lions, J. L., Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires (1969)
[20] Liu K., On stability for a class of semilinear stochastic evolution equations, Stochastic Process. Appl. 70 (1997), 219-241.; Liu, K., On stability for a class of semilinear stochastic evolution equations, Stochastic Process. Appl., 70, 219-241 (1997) · Zbl 0911.60049
[21] Liu K., Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 135, Chapman & Hall/CRC, Boca Raton, 2006.; Liu, K., Stability of Infinite Dimensional Stochastic Differential Equations with Applications (2006) · Zbl 1085.60003
[22] Pardoux E., Équations aux Dérivées Partielles Stochastiques non Linéaire Monotones, Ph.D. thesis, University Paris XI, Paris, 1975.; Pardoux, E., Équations aux Dérivées Partielles Stochastiques non Linéaire Monotones (1975) · Zbl 0363.60041
[23] Scheutzow M., Stabilization and destabilization by noise in the plane, Stoch. Anal. Appl. 11 (1993), 97-113.; Scheutzow, M., Stabilization and destabilization by noise in the plane, Stoch. Anal. Appl., 11, 97-113 (1993) · Zbl 0766.60072
[24] Temam R., Navier-Stokes Equations: Theory and Numerical Analysis, 2nd ed., North-Holland, Amsterdam, 1979.; Temam, R., Navier-Stokes Equations: Theory and Numerical Analysis (1979) · Zbl 0426.35003
[25] Yang M. and Kloeden P. E., Random attractors for stochastic semi-linear degenerate parabolic equations, Nonlinear Anal. Real World Appl. 12 (2011), 2811-2821.; Yang, M.; Kloeden, P. E., Random attractors for stochastic semi-linear degenerate parabolic equations, Nonlinear Anal. Real World Appl., 12, 2811-2821 (2011) · Zbl 1222.35042
[26] Yin J., Li Y. and Zhao H., Random attractors for stochastic semi-linear degenerate parabolic equations with additive noise in \({L^q}\), Appl. Math. Comput. 225 (2013), 526-540.; Yin, J.; Li, Y.; Zhao, H., Random attractors for stochastic semi-linear degenerate parabolic equations with additive noise in \({L^q}\), Appl. Math. Comput., 225, 526-540 (2013) · Zbl 1334.37053
[27] Zhao W. Q., Regularity of random attractors for a degenerate parabolic equations driven by additive noises, Appl. Math. Comput. 239 (2014), 358-374.; Zhao, W. Q., Regularity of random attractors for a degenerate parabolic equations driven by additive noises, Appl. Math. Comput., 239, 358-374 (2014) · Zbl 1334.35456
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