# American Institute of Mathematical Sciences

November  2018, 23(9): 3915-3934. doi: 10.3934/dcdsb.2018117

## Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives

 a). College of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China b). Department of Mathematical Sciences, School of Physical Sciences, The University of Liverpool, Liverpool, L69 7ZL, UK

The author is grateful to the Tianjin Thousand Talents Plan for its financial support.

Received  April 2017 Published  November 2018 Early access  April 2018

In this work, we shall consider the existence and uniqueness of stationary solutions to stochastic partial functional differential equations with additive noise in which a neutral type of delay is explicitly presented. We are especially concerned about those delays appearing in both spatial and temporal derivative terms in which the coefficient operator under spatial variables may take the same form as the infinitesimal generator of the equation. We establish the stationary property of the neutral system under investigation by focusing on distributed delays. In the end, an illustrative example is analyzed to explain the theory in this work.

Citation: Kai Liu. Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3915-3934. doi: 10.3934/dcdsb.2018117
##### References:
 [1] A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Math., A. K. Peters, Wellesley, Massachusetts, 2005. [2] E. B. Davies, One Parameter Semigroups, Academic Press, New York, 1980. [3] G. Di Blasio, K. Kunisch and E. Sinestrari, $L^2$-regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives, J. Math. Anal. Appl., 102 (1984), 38-57.  doi: 10.1016/0022-247X(84)90200-2. [4] G. Di Blasio, K. Kunisch and E. Sinestrari, Stability for abstract linear functional differential equations, Israel J. Math., 50 (1985), 231-263.  doi: 10.1007/BF02761404. [5] J. Hale and S. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, New York, Springer-Verlag, Heidelberg/Berlin, 1993. [6] K. Ito and T. Tarn, A linear quadratic optimal control for neutral systems, Nonlinear Anal. TMA., 9 (1985), 699-727.  doi: 10.1016/0362-546X(85)90013-6. [7] J. Jeong, Stabilizability of retarded functional differential equation in Hilbert spaces, Osaka J. Math., 28 (1991), 347-365. [8] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. Ⅰ. Springer-Verlag, Berlin, New York, 1972. [9] K. Liu, Uniform $L^2$-stability in mean square of linear autonomous stochastic functional differential equations in Hilbert spaces, Stoch. Proc. Appl., 115 (2005), 1131-1165.  doi: 10.1016/j.spa.2005.02.006. [10] K. Liu, On stationarity of stochastic retarded linear equations with unbounded drift operators, Stoch. Anal. Appl., 34 (2016), 547-572. [11] K. Liu, Norm continuity of solution semigroups of a class of neutral functional differential equations with distributed delay, Applied. Math. Letters., 69 (2017), 35-41.  doi: 10.1016/j.aml.2017.01.010. [12] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Math., 1905, Springer-Verlag, New York, 2007. [13] H. Tanabe, Equations of Evolution, Pitman, New York, 1979. [14] H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Dekker, New York, 1997.

show all references

The author is grateful to the Tianjin Thousand Talents Plan for its financial support.

##### References:
 [1] A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Math., A. K. Peters, Wellesley, Massachusetts, 2005. [2] E. B. Davies, One Parameter Semigroups, Academic Press, New York, 1980. [3] G. Di Blasio, K. Kunisch and E. Sinestrari, $L^2$-regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives, J. Math. Anal. Appl., 102 (1984), 38-57.  doi: 10.1016/0022-247X(84)90200-2. [4] G. Di Blasio, K. Kunisch and E. Sinestrari, Stability for abstract linear functional differential equations, Israel J. Math., 50 (1985), 231-263.  doi: 10.1007/BF02761404. [5] J. Hale and S. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, New York, Springer-Verlag, Heidelberg/Berlin, 1993. [6] K. Ito and T. Tarn, A linear quadratic optimal control for neutral systems, Nonlinear Anal. TMA., 9 (1985), 699-727.  doi: 10.1016/0362-546X(85)90013-6. [7] J. Jeong, Stabilizability of retarded functional differential equation in Hilbert spaces, Osaka J. Math., 28 (1991), 347-365. [8] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. Ⅰ. Springer-Verlag, Berlin, New York, 1972. [9] K. Liu, Uniform $L^2$-stability in mean square of linear autonomous stochastic functional differential equations in Hilbert spaces, Stoch. Proc. Appl., 115 (2005), 1131-1165.  doi: 10.1016/j.spa.2005.02.006. [10] K. Liu, On stationarity of stochastic retarded linear equations with unbounded drift operators, Stoch. Anal. Appl., 34 (2016), 547-572. [11] K. Liu, Norm continuity of solution semigroups of a class of neutral functional differential equations with distributed delay, Applied. Math. Letters., 69 (2017), 35-41.  doi: 10.1016/j.aml.2017.01.010. [12] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Math., 1905, Springer-Verlag, New York, 2007. [13] H. Tanabe, Equations of Evolution, Pitman, New York, 1979. [14] H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Dekker, New York, 1997.
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