The Existence of Stationary Solution for Nonlinear Random Reaction-Diffusion Equation in Banach Spaces

Ms. Sofije Hoxha; Fejzi Kolaneci

doi:https://doi.org/10.14445/22315373/IJMTT-V66I2P507

International Journal of Mathematics Trends and Technology

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Volume 66 | Issue 2 | Year 2020 | Article Id. IJMTT-V66I2P507 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I2P507

The Existence of Stationary Solution for Nonlinear Random Reaction-Diffusion Equation in Banach Spaces

Ms. Sofije Hoxha, Fejzi Kolaneci

Citation :

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

Abstract

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.

Keywords

random reaction-diffusion problem, real noise stationary solution

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