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Effects of the noise level on nonlinear stochastic fractional heat equations
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China |
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 $\frac{2\alpha}{\alpha-1}$.
References:
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E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D thesis, Eindhoven University of Technology, 2001. |
| [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. |
| [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. 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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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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