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October  2019, 24(10): 5437-5460. doi: 10.3934/dcdsb.2019065

Effects of the noise level on nonlinear stochastic fractional heat equations

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

Received  June 2018 Published  October 2019 Early access  April 2019

We consider the stochastic fractional heat equation $\partial_{t}u=\triangle^{\alpha/2}u+\lambda\sigma(u)\dot{w}$ on $[0,L]$ with Dirichlet boundary conditions, where $\dot{w}$ denotes the space-time white noise. For any $\lambda>0$, we prove that the $p$th moment of $\sup_{x\in [0,L]}|u(t,x)|$ grows at most exponentially. If $\lambda$ is small, we prove that the $p$th moment of $\sup_{x\in [0,L]}|u(t,x)|$ is exponentially stable. At last, we obtain the noise excitation index of $p$th energy of $u(t,x)$ is $\frac{2\alpha}{\alpha-1}$.

Citation: Kexue Li. Effects of the noise level on nonlinear stochastic fractional heat equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5437-5460. doi: 10.3934/dcdsb.2019065
References:
[1]

E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D thesis, Eindhoven University of Technology, 2001.

[2]

K. BogdanT. Grzywny and M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Probab., 38 (2010), 1901-1923.  doi: 10.1214/10-AOP532.

[3]

K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Commun. Math. Phys., 271 (2007), 179-198.  doi: 10.1007/s00220-006-0178-y.

[4]

Z.-Q. ChenP. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1307-1329.  doi: 10.4171/JEMS/231.

[5]

Z.-Q. ChenM. M. Meerschaert and E. Nane, Space-time fractional diffusion on bounded domains, J. Math. Anal. Appl., 393 (2012), 479-488.  doi: 10.1016/j.jmaa.2012.04.032.

[6]

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

[7]

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.

[8]

M. Foondun and E. Nualart, On the behavior of stochastic heat equations on bounded domains, ALEA, Lat. Am. J. Probab. Math. Stat., 12 (2015), 551-571. 

[9]

M. FoondunK. Tian and W. Liu, On some properties of a class of fractional stochastic heat equations, J. Theor. Probab., 30 (2017), 1310-1333.  doi: 10.1007/s10959-016-0684-6.

[10]

P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Cambridge University Press, Cambridge studies in advanced mathematics vol. 120, Cambridge 2010. doi: 10.1017/CBO9780511845079.

[11]

A. M. GarsiaE. Rodemich and H. Rumsey Jr, A real variable lemma and the continuity of paths of some Gaussian processes, Indiana Univ. Math. J., 20 (1970), 565-578.  doi: 10.1512/iumj.1971.20.20046.

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, Berlin, 1981.

[13]

T. Jakubowski and G. Serafin, Stable estimates for source solution of critical fractal Burgers equation, Nonlinear. Anal., 130 (2016), 396-407.  doi: 10.1016/j.na.2015.10.016.

[14]

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

[15]

D. Khoshnevisan and K. Kim, Nonlinear Noise Excitation of intermittent stochastic pdes and the topology of LCA groups, Ann. Probab., 43 (2015), 1944-1991.  doi: 10.1214/14-AOP925.

[16] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. 
[17] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970. 
[18]

J. B. Walsh, An introduction to stochastic partial differential equations, Lecture Notes in Math., Springer, Berlin, 1180 (1986), 265–439. doi: 10.1007/BFb0074920.

[19]

F. Wang and X. Zhang, Heat kernel for fractional diffusion operators with perturbations, Forum Math, 27 (2015), 973-994.  doi: 10.1515/forum-2012-0074.

[20]

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

show all references

References:
[1]

E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D thesis, Eindhoven University of Technology, 2001.

[2]

K. BogdanT. Grzywny and M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Probab., 38 (2010), 1901-1923.  doi: 10.1214/10-AOP532.

[3]

K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Commun. Math. Phys., 271 (2007), 179-198.  doi: 10.1007/s00220-006-0178-y.

[4]

Z.-Q. ChenP. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1307-1329.  doi: 10.4171/JEMS/231.

[5]

Z.-Q. ChenM. M. Meerschaert and E. Nane, Space-time fractional diffusion on bounded domains, J. Math. Anal. Appl., 393 (2012), 479-488.  doi: 10.1016/j.jmaa.2012.04.032.

[6]

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

[7]

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.

[8]

M. Foondun and E. Nualart, On the behavior of stochastic heat equations on bounded domains, ALEA, Lat. Am. J. Probab. Math. Stat., 12 (2015), 551-571. 

[9]

M. FoondunK. Tian and W. Liu, On some properties of a class of fractional stochastic heat equations, J. Theor. Probab., 30 (2017), 1310-1333.  doi: 10.1007/s10959-016-0684-6.

[10]

P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Cambridge University Press, Cambridge studies in advanced mathematics vol. 120, Cambridge 2010. doi: 10.1017/CBO9780511845079.

[11]

A. M. GarsiaE. Rodemich and H. Rumsey Jr, A real variable lemma and the continuity of paths of some Gaussian processes, Indiana Univ. Math. J., 20 (1970), 565-578.  doi: 10.1512/iumj.1971.20.20046.

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, Berlin, 1981.

[13]

T. Jakubowski and G. Serafin, Stable estimates for source solution of critical fractal Burgers equation, Nonlinear. Anal., 130 (2016), 396-407.  doi: 10.1016/j.na.2015.10.016.

[14]

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

[15]

D. Khoshnevisan and K. Kim, Nonlinear Noise Excitation of intermittent stochastic pdes and the topology of LCA groups, Ann. Probab., 43 (2015), 1944-1991.  doi: 10.1214/14-AOP925.

[16] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. 
[17] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970. 
[18]

J. B. Walsh, An introduction to stochastic partial differential equations, Lecture Notes in Math., Springer, Berlin, 1180 (1986), 265–439. doi: 10.1007/BFb0074920.

[19]

F. Wang and X. Zhang, Heat kernel for fractional diffusion operators with perturbations, Forum Math, 27 (2015), 973-994.  doi: 10.1515/forum-2012-0074.

[20]

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

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