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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

Kexue Li

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

Received  June 2018 Published  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 $\frac{2\alpha}{\alpha-1}$.

Keywords: Fractional heat kernel, stochastic fractional heat equations, Mittag-Leffler function, excitation index.
Mathematics Subject Classification: Primary: 60H15, 35K05.
Citation: Kexue Li. Effects of the noise level on nonlinear stochastic fractional heat equations. Discrete & 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.  Google Scholar

[2]

K. Bogdan, T. 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.  Google Scholar

[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.  Google Scholar

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Z.-Q. Chen, P. 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.  Google Scholar

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Z.-Q. Chen, M. 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.  Google Scholar

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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.  Google Scholar

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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

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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.   Google Scholar

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M. Foondun, K. 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.  Google Scholar

[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.  Google Scholar

[11]

A. M. Garsia, E. 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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[2]

K. Bogdan, T. 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.  Google Scholar

[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.  Google Scholar

[4]

Z.-Q. Chen, P. 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.  Google Scholar

[5]

Z.-Q. Chen, M. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[9]

M. Foondun, K. 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.  Google Scholar

[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.  Google Scholar

[11]

A. M. Garsia, E. 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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

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