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May  2020, 25(5): 1715-1728. doi: 10.3934/dcdsb.2019248

The effect of noise intensity on parabolic equations

Guangying Lv 1, , Hongjun Gao 2, , Jinlong Wei 3,, and  Jiang-Lun Wu 4,

1. 

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

Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing 210023, China
3. 

School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China
4. 

Department of Mathematics, Swansea University, Swansea SA2 8PP, UK

* Corresponding author: Jinlong Wei

Received  May 2019 Revised  July 2019 Published  November 2019

Fund Project: This work is partially supported by China NSF Grant Nos. 11771123, 11531006, 11501577, PAPD of Jiangsu Higher Education Institutions, Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application, and China Postdoctoral Science Foundation No. 2016M600427 and No. 2017T100385, and Postdoctoral Science Foundation of Jiangsu Province (Grant No. 1601141B).

In this paper, the effect of noise intensity on parabolic equations is considered. We focus on the effect of noise on the energy solutions of stochastic parabolic equations. By utilising Ito's formula and the energy estimate method, we obtain excitation indices of the solution $ u $ at time $ t $. Furthermore, we improve existing results by introducing a simple method to verify the existing results in the literature.

Keywords: Itô's formula, stochastic parabolic equation, energy estimate method.
Mathematics Subject Classification: Primary: 60H15, 60H40; Secondary: 35K20.
Citation: Guangying Lv, Hongjun Gao, Jinlong Wei, Jiang-Lun Wu. The effect of noise intensity on parabolic equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1715-1728. doi: 10.3934/dcdsb.2019248
References:
[1]

G. K. Batchelor and A. Townsend, The nature of turbulent flow at large wave numbers, Proc. Royal Society A, 199 (1949), 238–255. Google Scholar

[2]

P.-L. Chow, Stochastic Partial Differential Equations, Applied Mathematics and Nonlinear Science Series, Chapman Hall/CRC, Boca Raton, FL, 2007. doi: 10.1201/9781420010305.  Google Scholar

[3]

P.-L. Chow, Unbounded positive solutions of nonlinear parabolic Itô equations, Commun. Stoch. Anal., 3 (2009), 211-222.  doi: 10.31390/cosa.3.2.04.  Google Scholar

[4]

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.  Google Scholar

[5] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[6]

R. C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s, Electron. J. Probab., 4 (1999), 1-29.  doi: 10.1214/EJP.v4-43.  Google Scholar

[7]

R. C. Dalang, D. Khoshnevisan and T. Zhang, Global solutions to stochastic reaction-diffusion equations with super-linear drift and multiplicative noise, Ann. Probab., 47 (2019), 519-559.  doi: 10.1214/18-AOP1270.  Google Scholar

[8]

F. Delarue, F. Flandoli and D. Vincenzi, Noise prevents collapse of Vlasov-Poisson point charges, Comm. Pure Appl. Math., 67 (2014), 1700–1736. doi: 10.1002/cpa.21476.  Google Scholar

[9]

M. Dozzi and J. L$\acute{o}$pez-Mimbela, Finite-time blowup and existence of global positive solutions of a semi-linear SPDE, Stochastic Process. Appl., 120 (2010), 767–776. doi: 10.1016/j.spa.2009.12.003.  Google Scholar

[10]

H. W. Emmons, The laminar-turbulent transition in a boundary layer - Part 1, J. Aeronaut. Sci., 18 (1951), 490–498. doi: 10.2514/8.2010.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

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M. Foondun and M. Joseph, Remarks on non-linear noise excitability of some stochastic heat equations, Stochastic Process. Appl., 124 (2014), 3429–3440. doi: 10.1016/j.spa.2014.04.015.  Google Scholar

[14]

M. Foondun and D. Khoshnevisan, Intermittence and nonlinear parabolic stochastic partial differential equations, Electron. J. Probab., 14 (2009), 548-568.  doi: 10.1214/EJP.v14-614.  Google Scholar

[15]

M. Foondun, W. Liu and K. Tian, On some properties of a class of fractional stochastic heat equations, J. Theor. Probab., 30 (2016), 1310-1333.  doi: 10.1007/s10959-016-0684-6.  Google Scholar

[16]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, 224, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[17]

E. Hsu, Y. Wang and Z. Wang, Stochastic De Giorgi iteration and regularity of stochastic partial differential equations, Ann. Probab., 45 (2017), 2855-2866.  doi: 10.1214/16-AOP1126.  Google Scholar

[18]

D. Khoshnevisan and K. Kim, Non-linear noise excitation of intermittent stochastic PDEs and the topology of LCA group, Ann. Probab., 43 (2015), 1944–1991. doi: 10.1214/14-AOP925.  Google Scholar

[19]

D. Khoshnevisan and K. Kim, Non-linear noise excitation and intermittency under high disorder, Proc. Amer. Math. Soc., 143 (2015), 4073–4083. doi: 10.1090/S0002-9939-2015-12517-8.  Google Scholar

[20]

W. Liu, Well-posedness of stochastic partial differential equations with Lyapunov condition, J. Differential Equations, 255 (2013), 572-592.  doi: 10.1016/j.jde.2013.04.021.  Google Scholar

[21]

W. Liu and M. R{ö}ckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar

[22]

G. Y. 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.  Google Scholar

[23]

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

[24] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with LÉvy Noise (An Evolution Equation Approach), Encyclopedia of Mathematics and its Applications, 113, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511721373.  Google Scholar
[25] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge, 1999.   Google Scholar
[26]

T. Taniguchi, The existence and uniqueness of energy solutions to local non-Lipschitz stochastic evolution equations, J. Math. Anal. Appl., 360 (2009), 245-253.  doi: 10.1016/j.jmaa.2009.06.007.  Google Scholar

[27]

H. C. Tuckwell, Stochastic process in the neurosciences, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA. doi: 10.1137/1.9781611970159.  Google Scholar

[28]

B. Xie, Some effects of the noise intensity upon non-linear stochastic heat equations on $[0, 1]$, Stochastic Process. Appl., 126 (2016), 1184-1205.  doi: 10.1016/j.spa.2015.10.014.  Google Scholar

[29]

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.  Google Scholar

show all references

References:
[1]

G. K. Batchelor and A. Townsend, The nature of turbulent flow at large wave numbers, Proc. Royal Society A, 199 (1949), 238–255. Google Scholar

[2]

P.-L. Chow, Stochastic Partial Differential Equations, Applied Mathematics and Nonlinear Science Series, Chapman Hall/CRC, Boca Raton, FL, 2007. doi: 10.1201/9781420010305.  Google Scholar

[3]

P.-L. Chow, Unbounded positive solutions of nonlinear parabolic Itô equations, Commun. Stoch. Anal., 3 (2009), 211-222.  doi: 10.31390/cosa.3.2.04.  Google Scholar

[4]

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.  Google Scholar

[5] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[6]

R. C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s, Electron. J. Probab., 4 (1999), 1-29.  doi: 10.1214/EJP.v4-43.  Google Scholar

[7]

R. C. Dalang, D. Khoshnevisan and T. Zhang, Global solutions to stochastic reaction-diffusion equations with super-linear drift and multiplicative noise, Ann. Probab., 47 (2019), 519-559.  doi: 10.1214/18-AOP1270.  Google Scholar

[8]

F. Delarue, F. Flandoli and D. Vincenzi, Noise prevents collapse of Vlasov-Poisson point charges, Comm. Pure Appl. Math., 67 (2014), 1700–1736. doi: 10.1002/cpa.21476.  Google Scholar

[9]

M. Dozzi and J. L$\acute{o}$pez-Mimbela, Finite-time blowup and existence of global positive solutions of a semi-linear SPDE, Stochastic Process. Appl., 120 (2010), 767–776. doi: 10.1016/j.spa.2009.12.003.  Google Scholar

[10]

H. W. Emmons, The laminar-turbulent transition in a boundary layer - Part 1, J. Aeronaut. Sci., 18 (1951), 490–498. doi: 10.2514/8.2010.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

M. Foondun and M. Joseph, Remarks on non-linear noise excitability of some stochastic heat equations, Stochastic Process. Appl., 124 (2014), 3429–3440. doi: 10.1016/j.spa.2014.04.015.  Google Scholar

[14]

M. Foondun and D. Khoshnevisan, Intermittence and nonlinear parabolic stochastic partial differential equations, Electron. J. Probab., 14 (2009), 548-568.  doi: 10.1214/EJP.v14-614.  Google Scholar

[15]

M. Foondun, W. Liu and K. Tian, On some properties of a class of fractional stochastic heat equations, J. Theor. Probab., 30 (2016), 1310-1333.  doi: 10.1007/s10959-016-0684-6.  Google Scholar

[16]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, 224, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[17]

E. Hsu, Y. Wang and Z. Wang, Stochastic De Giorgi iteration and regularity of stochastic partial differential equations, Ann. Probab., 45 (2017), 2855-2866.  doi: 10.1214/16-AOP1126.  Google Scholar

[18]

D. Khoshnevisan and K. Kim, Non-linear noise excitation of intermittent stochastic PDEs and the topology of LCA group, Ann. Probab., 43 (2015), 1944–1991. doi: 10.1214/14-AOP925.  Google Scholar

[19]

D. Khoshnevisan and K. Kim, Non-linear noise excitation and intermittency under high disorder, Proc. Amer. Math. Soc., 143 (2015), 4073–4083. doi: 10.1090/S0002-9939-2015-12517-8.  Google Scholar

[20]

W. Liu, Well-posedness of stochastic partial differential equations with Lyapunov condition, J. Differential Equations, 255 (2013), 572-592.  doi: 10.1016/j.jde.2013.04.021.  Google Scholar

[21]

W. Liu and M. R{ö}ckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar

[22]

G. Y. 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.  Google Scholar

[23]

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

[24] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with LÉvy Noise (An Evolution Equation Approach), Encyclopedia of Mathematics and its Applications, 113, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511721373.  Google Scholar
[25] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge, 1999.   Google Scholar
[26]

T. Taniguchi, The existence and uniqueness of energy solutions to local non-Lipschitz stochastic evolution equations, J. Math. Anal. Appl., 360 (2009), 245-253.  doi: 10.1016/j.jmaa.2009.06.007.  Google Scholar

[27]

H. C. Tuckwell, Stochastic process in the neurosciences, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA. doi: 10.1137/1.9781611970159.  Google Scholar

[28]

B. Xie, Some effects of the noise intensity upon non-linear stochastic heat equations on $[0, 1]$, Stochastic Process. Appl., 126 (2016), 1184-1205.  doi: 10.1016/j.spa.2015.10.014.  Google Scholar

[29]

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.  Google Scholar

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