October 2017, 37(10): 5105-5125. doi: 10.3934/dcds.2017221

Explosive solutions of parabolic stochastic partial differential equations with lévy noise

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

* Corresponding author

Received  August 2016 Revised  May 2017 Published  June 2017

In this paper, we study the explosive solutions to a class of parbolic stochastic semilinear differential equations driven by a Lévy type noise. The sufficient conditions are presented to guarantee the existence of a unique positive solution of the stochastic partial differential equation under investigation. Moreover, we show that positive solutions will blow up in finite time in mean Lp-norm sense, provided that the initial data, the nonlinear term and the multiplicative noise satisfies some conditions. Several examples are presented to illustrate the theory. Finally, we establish a global existence theorem based on a Lyapunov functional and prove that a stochastic Allen-Cahn equation driven by Lévy noise has a global solution.

Citation: Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221
References:
[1]

D. Applebaum, Lévy Processes and Stochastic Calculus 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.

[2]

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

[3]

J. F. Bonder and P. Groisman, Time-space white noise eliminates global solutions in reaction-diffusion equations, Physica D, 238 (2009), 209-215. doi: 10.1016/j.physd.2008.09.005.

[4]

Z. Brźeniak and J. Zabczyk, Regularity of Ornstein-Uhlenbeck processes driven by a Lévy white noise, Potential Anal, 32 (2010), 153-188. doi: 10.1007/s11118-009-9149-1.

[5]

P.-L. Chow, Explosive solutions of stochastic reaction-diffusion equations in mean Lp-norm, J. Differential Equations, 250 (2011), 2567-2580. doi: 10.1016/j.jde.2010.11.008.

[6]

P.-L. Chow and K. Liu, Positivity and explosion in mean Lp-norm of stochastic functional parabolic equations of retarded type, Stoch. Proc. Appl, 122 (2012), 1709-1729. doi: 10.1016/j.spa.2012.01.012.

[7]

P. -L. Chow, Stochastic Partial Differential Equations Second edition. Advances in Applied Mathematics. CRC Press, Boca Raton, FL, 2015.

[8]

P.-L. Chow, Unbounded positive solutions of nonlinear parabolic Itô equations, Commun. Stoch. Anal, 3 (2009), 211-222.

[9]

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

[10]

Z. Dong, On the uniqueness of invariant measure of the Burgers equation driven by Lévy processes, J. Theor. Probab, 21 (2008), 322-335. doi: 10.1007/s10959-008-0143-0.

[11]

M. Dozzi and J. A. López-Mimbela, Finite-time blowup and existence of global positive solutions of a semi-linear SPDE, Stoch. Proc. Appl, 120 (2010), 767-776. doi: 10.1016/j.spa.2009.12.003.

[12]

L. C. Evans, Partial Differential Equations 2nd edition, in Graduate Studies in Math., vol. 19, AMS, Providence, Rhode Island, 1998. doi: 10.1090/gsm/019.

[13]

H. Fujita, On the blowing up of solutions of the Cauchy problen for ut = ∆u + u1+α, J. Fac. Sci. Univ. Tokyo, Sect. 1, 13 (1966), 109-124.

[14]

H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, Proc. Symp. Pure Math, AMS, 18 (1970), 105-113.

[15]

V. A. Galaktionov and J. L. Vá, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dyn. Syst, 8 (2002), 399-433. doi: 10.3934/dcds.2002.8.399.

[16]

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

[17]

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

[18]

Y. LiX. Sun and Y. Xie, Fokker-Planck equations and maximal dissipativity for Kolmogorov operators for SPDE driven by Lévy noise, Potential Anal, 38 (2013), 381-396. doi: 10.1007/s11118-012-9277-x.

[19]

G. 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.

[20]

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

[21]

C. Mueller, The critical parameter for the heat equation with a noise term to blow up in finite time, Ann. Probab, 25 (1997), 133-152. doi: 10.1214/aop/1024404282.

[22]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511721373.

[23]

M. Röckner and T. Zhang, Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles, Potential Anal, 26 (2007), 255-279. doi: 10.1007/s11118-006-9035-z.

[24]

T. Shen and J. Huang, Well-posedness of the stochastic fractional Boussinesq equation with Lévy noise, Stoch. Anal. Appl, 33 (2015), 1092-1114. doi: 10.1080/07362994.2015.1089410.

[25]

F.-Y. WangL. Xu and X. Zhang, Gradient estimates for SDEs driven by multiplicative Lévy noise, J. Funct. Anal., 269 (2015), 3195-3219. doi: 10.1016/j.jfa.2015.09.007.

[26]

B. Xie, Uniqueness of invariant measures of infinite dimensional stochastic differential equations driven by Lévy noise, Potential Anal., 36 (2012), 35-66. doi: 10.1007/s11118-011-9220-6.

[27]

M. Yang, A parabolic Triebel-Lizorkin estimates for the fractional Laplacian operator, Proc. Amer. Math. Soc., 143 (2015), 2571-2578. doi: 10.1090/S0002-9939-2015-12523-3.

show all references

References:
[1]

D. Applebaum, Lévy Processes and Stochastic Calculus 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.

[2]

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

[3]

J. F. Bonder and P. Groisman, Time-space white noise eliminates global solutions in reaction-diffusion equations, Physica D, 238 (2009), 209-215. doi: 10.1016/j.physd.2008.09.005.

[4]

Z. Brźeniak and J. Zabczyk, Regularity of Ornstein-Uhlenbeck processes driven by a Lévy white noise, Potential Anal, 32 (2010), 153-188. doi: 10.1007/s11118-009-9149-1.

[5]

P.-L. Chow, Explosive solutions of stochastic reaction-diffusion equations in mean Lp-norm, J. Differential Equations, 250 (2011), 2567-2580. doi: 10.1016/j.jde.2010.11.008.

[6]

P.-L. Chow and K. Liu, Positivity and explosion in mean Lp-norm of stochastic functional parabolic equations of retarded type, Stoch. Proc. Appl, 122 (2012), 1709-1729. doi: 10.1016/j.spa.2012.01.012.

[7]

P. -L. Chow, Stochastic Partial Differential Equations Second edition. Advances in Applied Mathematics. CRC Press, Boca Raton, FL, 2015.

[8]

P.-L. Chow, Unbounded positive solutions of nonlinear parabolic Itô equations, Commun. Stoch. Anal, 3 (2009), 211-222.

[9]

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

[10]

Z. Dong, On the uniqueness of invariant measure of the Burgers equation driven by Lévy processes, J. Theor. Probab, 21 (2008), 322-335. doi: 10.1007/s10959-008-0143-0.

[11]

M. Dozzi and J. A. López-Mimbela, Finite-time blowup and existence of global positive solutions of a semi-linear SPDE, Stoch. Proc. Appl, 120 (2010), 767-776. doi: 10.1016/j.spa.2009.12.003.

[12]

L. C. Evans, Partial Differential Equations 2nd edition, in Graduate Studies in Math., vol. 19, AMS, Providence, Rhode Island, 1998. doi: 10.1090/gsm/019.

[13]

H. Fujita, On the blowing up of solutions of the Cauchy problen for ut = ∆u + u1+α, J. Fac. Sci. Univ. Tokyo, Sect. 1, 13 (1966), 109-124.

[14]

H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, Proc. Symp. Pure Math, AMS, 18 (1970), 105-113.

[15]

V. A. Galaktionov and J. L. Vá, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dyn. Syst, 8 (2002), 399-433. doi: 10.3934/dcds.2002.8.399.

[16]

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

[17]

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

[18]

Y. LiX. Sun and Y. Xie, Fokker-Planck equations and maximal dissipativity for Kolmogorov operators for SPDE driven by Lévy noise, Potential Anal, 38 (2013), 381-396. doi: 10.1007/s11118-012-9277-x.

[19]

G. 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.

[20]

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

[21]

C. Mueller, The critical parameter for the heat equation with a noise term to blow up in finite time, Ann. Probab, 25 (1997), 133-152. doi: 10.1214/aop/1024404282.

[22]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511721373.

[23]

M. Röckner and T. Zhang, Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles, Potential Anal, 26 (2007), 255-279. doi: 10.1007/s11118-006-9035-z.

[24]

T. Shen and J. Huang, Well-posedness of the stochastic fractional Boussinesq equation with Lévy noise, Stoch. Anal. Appl, 33 (2015), 1092-1114. doi: 10.1080/07362994.2015.1089410.

[25]

F.-Y. WangL. Xu and X. Zhang, Gradient estimates for SDEs driven by multiplicative Lévy noise, J. Funct. Anal., 269 (2015), 3195-3219. doi: 10.1016/j.jfa.2015.09.007.

[26]

B. Xie, Uniqueness of invariant measures of infinite dimensional stochastic differential equations driven by Lévy noise, Potential Anal., 36 (2012), 35-66. doi: 10.1007/s11118-011-9220-6.

[27]

M. Yang, A parabolic Triebel-Lizorkin estimates for the fractional Laplacian operator, Proc. Amer. Math. Soc., 143 (2015), 2571-2578. doi: 10.1090/S0002-9939-2015-12523-3.

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