International Journal of Mathematics Trends and Technology
Volume 66 | Issue 2 | Year 2020 | Article Id. IJMTT-V66I2P507 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I2P507
We study a nonlinear random reaction-diffusion problem in abstract Banach spaces, driven by a real noise, with random diffusion coefficient and random initial condition. The reaction-diffusion equation belongs to the class of parabolic stochastic partial differential equations. Given a Gelfand triplet V ⊂ H = H' ⊂ V' with dense embeddings .Let A (t) be a family of nonlinear random operations, acting from V to V' , t ∈ R+, which satisfies the following assumptions: strong measurability, continuity, monotony, and coercivity. We assume that the initial condition is an element of Hilbert space. We construct a suitable stochastic basis and define the solution of reaction-diffusion problem in the weak sense. We define the stationary process in abstract Banach spaces in the strong sense of Doob-Rozanov. That is, the probability density function of the stochastic process is independent of time shift. In other words, we define the invariant measure for random dynamical system, associated with random reaction-diffusion problem. We prove the existence of an invariant measure and the existence of a stationary solution for nonlinear random reaction-diffusion problem. The obtained theoretical results have several applications in Quantum Physics, Biology, Medicine, and Economic Sciences. Especially, we can study the existence of stationary solution for the stochastic models of tumor growth.
[1] L. Arnold, W.Kliemann.Qualitative Theory of Stochastic Systems, in: Probability Anallysis and Related Topics, A.T.Bharucha-Reid ed., vol. III, Academics Pres, New York, 1983, 1-77.
[2] S.K Chung T.A. Austin Modeling Saturated-Unsaturated Water Flow in Soil. J. Irrig. Drain. Engin. 113 (2), 1987, 233-250.
[3] J.R Cordova, R. L. Bras. Physically Based Probabilistic Models of Infiltration, Soil Moisiture and Actual Evapotranspiration. Water Resources Research 17 (1), 1981, 93-106.
[4] G. DaPrato, J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge, 1992.
[5] F. Flandoli, D. Gatarek. Martingale and Stationary Solutions for Stochastic Naiver- Stokes Equations. Probab. Theory Relat. Fields, 102, 1995, 367-391.
[6] F. Flandoli, F. Kolaneci. Invariant Measures for Random Differential Equations in Infinite Dimensions. Preprints di Matematica, n.26, 1994, Scuola Normale Superiore, Pisa.
[7] R.A. Freeze. A Stochastic Conceptual Analysis of One-Dimensional Water Flow in Non-uniform Homogeneous Media. Water Resources Research, 1975, 725-741.
[8] D. Gatarek. A. Note on Nonlinear Stochastic Equations in Hilbert Spaces. Stad. and Prob. Letters 17, 1993, 387-394.
[9] N. Ikeda, S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland, Kodansha, 1981.
[10] F. Kolaneci. The Studay of a Stochastic Model of the Water Flow in Soil. Bull. Shk. Nat. 4 (1988), Tirana, 35-43.
[11] N.V. Krylov, B.L. Rozowski. Stochastic Evolution Equations. J.Sov. Math. 16 (1981), 1233-1277.
[12] Ch.Kuttler, Reaction-Diffusion Equations with Applications, 2011 (personal communication).
[13] J.L. Lions. Quelques Methodes de Resolution des Problemes aux Limites non Lineares. Dunod, Paris, 1969.
[14] M. Metivier. Stochastic Partial Differential Equations in Infinite Dimensional Spaces. Quaderni Scuola Normale Superiore, Pisa, 1988.
[15] Y. Miyahara. Invariant Measures for Ultimately Bounded Stochastic Processes. Nagoya Math. J., 49, 1973, 149-153.
[16] J.Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, 2003.
[17] E. Pardoux. Equations aux Derivée Partilles Stochastiques Nonlineaires Monotones, These, Universite Paris XI, 1975.
[18] B. Sagar, C. Preller. Stochastic Model of Water Table under Drainage. J. Irrig. Drain. Engin. 106 (3), 1980, 189-202.
[19] P. Stengel. Utilisation de l’ Analyse des Systemes de Porosite pour la Caracterisation de l’Etat Physique du Sol in Situ. Annal. Agronom. 1, 1979, 27-49.
[20] J. Viot. Solutions Faibles d’Equations aux Derivees Partielles Stochastiques non Lineaires. These de Doctorat, Paris VI, 1976.
Ms. Sofije Hoxha, Fejzi Kolaneci, "The Existence of Stationary Solution for Nonlinear Random Reaction-Diffusion Equation in Banach Spaces," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 2, pp. 58-63, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I2P507
PDF 
Abstract 
Keywords 
References 
Citation