October  2019, 24(10): 5769-5784. doi: 10.3934/dcdsb.2019105

Impacts of noise on heat equations

1. 

Institute of Applied Mathematics, Henan University, Kaifeng, Henan 475001, China

2. 

Center for Applied Mathematics, Tianjin University, Tianjin 300072, China

3. 

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

Received  July 2018 Revised  December 2018 Published  June 2019

Fund Project: The first author was supported in part by NSFC of China grants 11771123, 11726628 and the second author was supported in part by NSFC of China grants 11531006. The authors are grateful to the referees for their valuable suggestions and comments on the original manuscript.

In this paper, we consider the impacts of noise on heat equations. Our results show that the noise can induce singularities (finite time blow up of solutions) and that the nonlinearity can prevent the singularities. Moreover, suitable noise can prevent the solution vanishing. Besides that, we obtain the solutions of some reaction-diffusion equations keep positive, included stochastic Burgers' equation.

Citation: Guangying Lv, Hongjun Gao. Impacts of noise on heat equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5769-5784. doi: 10.3934/dcdsb.2019105
References:
[1]

J. Bao and C. Yuan, Blow-up for stochastic reactin-diffusion equations with jumps, J Theor. Probab., 29 (2016), 617-631.  doi: 10.1007/s10959-014-0589-1.  Google Scholar

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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

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M. Dozzi and J. A. López-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

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H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t-\Delta u = u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 13 (1966), 109-124.   Google Scholar

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H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, 1970 Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, Ill., 1968), 1968,105–113 Amer. Math. Soc., Providence, R.I.  Google Scholar

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S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330.  doi: 10.1002/cpa.3160160307.  Google Scholar

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W. Liu and M. Röckner, SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal., 259 (2010), 2902-2922.  doi: 10.1016/j.jfa.2010.05.012.  Google Scholar

[12]

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

[13]

R. Manthey and T. Zausinger, Stochastic evolution equations in $L^{2\nu}_\rho$, Stochastics and Stochastic Report, 66 (1999), 37-85.  doi: 10.1080/17442509908834186.  Google Scholar

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C. Mueller, Long time existence for the heat equation with a noise term, Probab. Theory Related Fields, 90 (1991), 505-517.  doi: 10.1007/BF01192141.  Google Scholar

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C. Mueller and R. Sowers, Blowup for the heat equation with a noise term, Probab. Theory Related Fields, 97 (1993), 287-320.  doi: 10.1007/BF01195068.  Google Scholar

[16]

M. Niu and B. Xin, Impacts of Gaussian noises on the blow-up times of nonlinear stochastic partial differential equations, Nonlinear Analysis: Real World Applications, 13 (2012), 1346-1352.  doi: 10.1016/j.nonrwa.2011.10.011.  Google Scholar

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

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G. Da Prato and J. Zabczyk, Nonexplosion, boundedness and ergodicity for stochastic semilinear equations, J. Differential Equations, 98 (1992), 181-195.  doi: 10.1016/0022-0396(92)90111-Y.  Google Scholar

[19]

A. Samarskii, V. Galaktionov, S. Kurdyumov and S. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, New York, 1995. doi: 10.1515/9783110889864.535.  Google Scholar

[20]

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

show all references

References:
[1]

J. Bao and C. Yuan, Blow-up for stochastic reactin-diffusion equations with jumps, J Theor. Probab., 29 (2016), 617-631.  doi: 10.1007/s10959-014-0589-1.  Google Scholar

[2]

P.-L. Chow, Stochastic Partial Differential Equations, Chapman Hall/CRC Applied Mathematics and Nonlinear Science Series. Chapman Hall/CRC, Boca Raton, FL, 2007. x+281 pp. ISBN: 978-1-58488-443-9.  Google Scholar

[3]

P.-L. Chow, Unbounded positive solutions of nonlinear parabolic Itô equations, Communications on Stochastic Analysis, 3 (2009), 211-222.   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]

M. Dozzi and J. A. López-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

[6]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t-\Delta u = u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 13 (1966), 109-124.   Google Scholar

[7]

H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, 1970 Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, Ill., 1968), 1968,105–113 Amer. Math. Soc., Providence, R.I.  Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Ed., Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[9]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad. Ser. A Math., 49 (1973), 503-505.  doi: 10.3792/pja/1195519254.  Google Scholar

[10]

S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330.  doi: 10.1002/cpa.3160160307.  Google Scholar

[11]

W. Liu and M. Röckner, SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal., 259 (2010), 2902-2922.  doi: 10.1016/j.jfa.2010.05.012.  Google Scholar

[12]

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

[13]

R. Manthey and T. Zausinger, Stochastic evolution equations in $L^{2\nu}_\rho$, Stochastics and Stochastic Report, 66 (1999), 37-85.  doi: 10.1080/17442509908834186.  Google Scholar

[14]

C. Mueller, Long time existence for the heat equation with a noise term, Probab. Theory Related Fields, 90 (1991), 505-517.  doi: 10.1007/BF01192141.  Google Scholar

[15]

C. Mueller and R. Sowers, Blowup for the heat equation with a noise term, Probab. Theory Related Fields, 97 (1993), 287-320.  doi: 10.1007/BF01195068.  Google Scholar

[16]

M. Niu and B. Xin, Impacts of Gaussian noises on the blow-up times of nonlinear stochastic partial differential equations, Nonlinear Analysis: Real World Applications, 13 (2012), 1346-1352.  doi: 10.1016/j.nonrwa.2011.10.011.  Google Scholar

[17]

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

[18]

G. Da Prato and J. Zabczyk, Nonexplosion, boundedness and ergodicity for stochastic semilinear equations, J. Differential Equations, 98 (1992), 181-195.  doi: 10.1016/0022-0396(92)90111-Y.  Google Scholar

[19]

A. Samarskii, V. Galaktionov, S. Kurdyumov and S. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, New York, 1995. doi: 10.1515/9783110889864.535.  Google Scholar

[20]

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

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