Asymptotic behavior for neutral stochastic partial differential equations with infinite delays

Jing Cui (Anhui Normal University)
Litan Yan (Donghua University)


This paper is concerned with the existence and asymptotic behavior of mild solutions to a class of non-linear neutral stochastic partial differential equations with infinite delays. By applying fixed point principle, we present sufficient conditions to ensure that the mild solutions are exponentially stable in $p$th-moment ($p\geq 2$) and almost surely exponentially stable. An example is provided to illustrate the effectiveness of the proposed result.

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Pages: 1-12

Publication Date: June 8, 2013

DOI: 10.1214/ECP.v18-2858


  • Albeverio, S.; Mandrekar, V.; Rüdiger, B. Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise. Stochastic Process. Appl. 119 (2009), no. 3, 835--863. MR2499860
  • Burton, T. A. Stability by fixed point theory or Liapunov theory: a comparison. Fixed Point Theory 4 (2003), no. 1, 15--32. MR2031819
  • Burton, T. A. Fixed points, stability, and exact linearization. Nonlinear Anal. 61 (2005), no. 5, 857--870. MR2130068
  • Burton, T. A.; Furumochi, Tetsuo. Asymptotic behavior of solutions of functional differential equations by fixed point theorems. Dynam. Systems Appl. 11 (2002), no. 4, 499--519. MR1946140
  • Burton, T. A.; Zhang, Bo. Fixed points and stability of an integral equation: nonuniqueness. Appl. Math. Lett. 17 (2004), no. 7, 839--846. MR2072844
  • Chow, Pao Liu. Stability of nonlinear stochastic-evolution equations. J. Math. Anal. Appl. 89 (1982), no. 2, 400--419. MR0677738
  • Caraballo, Tomás; Real, José. Partial differential equations with delayed random perturbations: existence, uniqueness, and stability of solutions. Stochastic Anal. Appl. 11 (1993), no. 5, 497--511. MR1243598
  • Caraballo, Tomás; Liu, Kai. Exponential stability of mild solutions of stochastic partial differential equations with delays. Stochastic Anal. Appl. 17 (1999), no. 5, 743--763. MR1714897
  • Cui, Jing; Yan, Litan; Sun, Xichao. Exponential stability for neutral stochastic partial differential equations with delays and Poisson jumps. Statist. Probab. Lett. 81 (2011), no. 12, 1970--1977. MR2845915
  • Hernández M, Eduardo; Henríquez, Hernán R.; dos Santos, José Paulo C. Existence results for abstract partial neutral integro-differential equation with unbounded delay. Electron. J. Qual. Theory Differ. Equ. 2009, No. 29, 23 pp. MR2501417
  • Kolmanovskiĭ, V.; Myshkis, A. Applied theory of functional-differential equations. Mathematics and its Applications (Soviet Series), 85. Kluwer Academic Publishers Group, Dordrecht, 1992. xvi+234 pp. ISBN: 0-7923-2013-1 MR1256486
  • Kuang, Yang. Delay differential equations with applications in population dynamics. Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993. xii+398 pp. ISBN: 0-12-427610-5 MR1218880
  • Liu, Kai; Truman, Aubrey. A note on almost sure exponential stability for stochastic partial functional differential equations. Statist. Probab. Lett. 50 (2000), no. 3, 273--278. MR1792306
  • Luo, Jiaowan. Fixed points and stability of neutral stochastic delay differential equations. J. Math. Anal. Appl. 334 (2007), no. 1, 431--440. MR2332567
  • Luo, Jiaowan. Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays. J. Math. Anal. Appl. 342 (2008), no. 2, 753--760. MR2433617
  • Luo, Jiaowan; Taniguchi, Takeshi. Fixed points and stability of stochastic neutral partial differential equations with infinite delays. Stoch. Anal. Appl. 27 (2009), no. 6, 1163--1173. MR2573454
  • Liu, Kai. Stability of infinite dimensional stochastic differential equations with applications. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 135. Chapman & Hall/CRC, Boca Raton, FL, 2006. xii+298 pp. ISBN: 978-1-58488-598-6; 1-58488-598-X MR2165651
  • Liu, Kai; Truman, Aubrey. Moment and almost sure Lyapunov exponents of mild solutions of stochastic evolution equations with variable delays via approximation approaches. J. Math. Kyoto Univ. 41 (2001), no. 4, 749--768. MR1891673
  • Nunziato, Jace W. On heat conduction in materials with memory. Quart. Appl. Math. 29 (1971), 187--204. MR0295683
  • Pazy, A. Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. viii+279 pp. ISBN: 0-387-90845-5 MR0710486
  • Da Prato, Giuseppe; Zabczyk, Jerzy. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. xviii+454 pp. ISBN: 0-521-38529-6 MR1207136
  • Taniguchi, Takeshi. The exponential stability for stochastic delay partial differential equations. J. Math. Anal. Appl. 331 (2007), no. 1, 191--205. MR2305998
  • Taniguchi, Takeshi; Luo, Jiaowan. The existence and asymptotic behaviour of mild solutions to stochastic evolution equations with infinite delays driven by Poisson jumps. Stoch. Dyn. 9 (2009), no. 2, 217--229. MR2531628
  • Wang, Feng-Yu; Zhang, Tu-Sheng. Gradient estimates for stochastic evolution equations with non-Lipschitz coefficients. J. Math. Anal. Appl. 365 (2010), no. 1, 1--11. MR2585069
  • Xu, Tiange; Zhang, Tusheng. White noise driven SPDEs with reflection: existence, uniqueness and large deviation principles. Stochastic Process. Appl. 119 (2009), no. 10, 3453--3470. MR2568282

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