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  October 2019 Early access  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 and 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.

[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.

[3]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[1]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221

[2]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209

[3]

Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100

[4]

Nobuyuki Kato, Norio Kikuchi. Campanato-type boundary estimates for Rothe's scheme to parabolic partial differential systems with constant coefficients. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 737-760. doi: 10.3934/dcds.2007.19.737

[5]

Leonid Shaikhet. Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3651-3657. doi: 10.3934/dcdsb.2020077

[6]

Shota Sato, Eiji Yanagida. Appearance of anomalous singularities in a semilinear parabolic equation. Communications on Pure and Applied Analysis, 2012, 11 (1) : 387-405. doi: 10.3934/cpaa.2012.11.387

[7]

Wojciech Kryszewski, Dorota Gabor, Jakub Siemianowski. The Krasnosel'skii formula for parabolic differential inclusions with state constraints. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 295-329. doi: 10.3934/dcdsb.2018021

[8]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[9]

Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199

[10]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[11]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

[12]

Gamaliel Blé, Carlos Cabrera. A generalization of Douady's formula. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6183-6188. doi: 10.3934/dcds.2017267

[13]

Avner Friedman, Harsh Vardhan Jain. A partial differential equation model of metastasized prostatic cancer. Mathematical Biosciences & Engineering, 2013, 10 (3) : 591-608. doi: 10.3934/mbe.2013.10.591

[14]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control and Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231

[15]

Xiaojun Li, Xiliang Li, Kening Lu. Random attractors for stochastic parabolic equations with additive noise in weighted spaces. Communications on Pure and Applied Analysis, 2018, 17 (3) : 729-749. doi: 10.3934/cpaa.2018038

[16]

Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103

[17]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135

[18]

Min Yang, Guanggan Chen. Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1565-1581. doi: 10.3934/dcdsb.2019240

[19]

Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021020

[20]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (152)
  • HTML views (279)
  • Cited by (0)

Other articles
by authors

[Back to Top]