# EXISTENCE AND UNIQUENESS RESULTS, THE MARKOVIAN PROPERTY OF SOLUTION FOR A NEUTRAL DELAY STOCHASTIC REACTION-DIFFUSION EQUATION IN ENTIRE SPACE

 Title EXISTENCE AND UNIQUENESS RESULTS, THE MARKOVIAN PROPERTY OF SOLUTION FOR A NEUTRAL DELAY STOCHASTIC REACTION-DIFFUSION EQUATION IN ENTIRE SPACE Publication Type Journal Article Year of Publication 2019 Authors KENZHEBAEV K.K., STANZHYTSKYI A.N., TSUKANOVA A.O. Journal Dynamic Systems and Applications Volume 28 Issue 1 Start Page 19 Pagination 28 Date Published 10/2018 ISSN 1056-2176 AMS Subject Classification 34K40, 34K50, 60H15 Abstract We establish results, concerning existence, uniqueness, and continuous dependence on an initial datum of a mild solution for neutral stochastic integro-differential equations with variable time delay of reaction-diffusion type. We also establish the Markovian property of this solution. Herewith our emphasis is on unbounded domain $\{x \in \mathbb{R}^{d}\}$. PDF https://acadsol.eu/dsa/articles/28/1/2.pdf DOI 10.12732/dsa.v28i1.2 Refereed Designation Refereed Full Text REFERENCES [1] V. Capasso, D. Fortunato, Stability results for semilinear evolution equations and their application to some reaction-diffusion problems, SIAM J. Math. Anal., 39 (1980), 37-47. [2] Ye. F. Carkov, Sluchajnye Vozmusheniya Differencial’no-Funkcionalnyh Uravneny [Random Perturbations of Functional Differential Equations] (in Russian), Zinatne, Riga, 1989. [3] G. Da Prato, J. Zabczyk, Ergodicity for infinite dimensional systems, London Mathematical Society Lecture Note Series, 229, Cambridge University Press, (1996). [4] G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, Vol. 45, Cambridge University Press, 1992. [5] D. A. Dawson, Stochastic Evolution Equations, Math. Biosci., 15 (1972), 287-316. [6] Y. Du, S. Hsu, On a nonlocal reaction-diffusion problem arising from the modeling of phytoplankton growth, SIAM J. Math. Anal., 42 (2010), 1305-1333. [7] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, R. I., 1998. [8] J. K. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York, 1977. [9] K. Ito, M. Nisio, On stationary solutions of a stochastic differential equation, 4 (1964), 1-75. [10] A. F. Ivanov, Y. I. Kazmerchuk, A. V. Swishchuk, Theory, stochastic stability and applications of stochastic delay differential equations: a survey of recent results, Diff. Equ. and Dynam. Systems, 11 (2003), 55-115. [11] V. B. Kolmanovskii, L. E. Shaikhet, Upravleniye Sistemami s Posledejstviyem [Control of Systems with After-Effect] (in Russian), Trans. Amer. Math. Soc., Providence, R. I., 157, 1996. [12] V. B. Kolmanovskii, V. M. Nosov, Ustojchivost’ i Periodicheskiye Rezhimy Reguliruyemykh Sistem s Posledejstviyem [Stability and Periodic Models of Control Systems with After-Effect] (in Russian), Nauka, Moscow, 1981. [13] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural’tseva, Linejnye i KvaziLinejnye Uravneniya Parabolicheskogo Tipa [Linear and Quasi-Linear Equations of Parabolic Type] (in Russian), Trans. Amer. Math. Soc., Providence, R. I., 23, 1968. [14] N. I. Mahmudov, Existence and uniqueness results for neutral SDEs in Hilbert spaces, Stochastic Anal. Appl., 24 (2006), 79-95. [15] X. Mao, Asymptotic properties of neutral stochastic differential delay equations, Stochastics Rep., 68 (2000), 273-295. [16] X. Mao, Exponential stability in mean square of neutral stochastic differential difference equations, Dynam. Contin. Discrete Impuls. Systems, 6 (1999), 569-586. [17] X.Mao, A. Rodkina, N. Koroleva, Razumikhin-type theorems for neutral stochastic functional differential equations, Funct. Differ. Equ., 5 (1998), 195-211. [18] H. P. McKean, Nagumo’s equation, Advances in Math., 4 (1970), 209-223.  [19] V. Volpert, Elliptic Partial Differential Equations, Monographs in Mathematics, Vol. 104, Birkh¨auser, Springer-Basel, 2014. [20] K. Yosida, Functional Analysis, Grundlehren derMathematischenWissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 123, Springer-Verlag, Berlin-New York, 1980.