# American Institue of Mathematical Sciences

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. [2] J. Bao, 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, 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, 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, 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, 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, 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. [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, 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. [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. Li, X. Sun, 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, 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. [23] M. Röckner, 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, 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. Wang, L. Xu, 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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##### References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus 2nd edition, Cambridge University Press, Cambridge, 2009. [2] J. Bao, 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, 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, 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, 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, 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, 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. [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, 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. [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. Li, X. Sun, 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, 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. [23] M. Röckner, 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, 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. Wang, L. Xu, 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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