Random attractors for semilinear reaction-diffusion equation with distribution derivatives and multiplicative noise on \(\mathbb{R}^{n}\)

Fadlallah Mustafa Mosa,Mohamed Y A Bakhet,Abdelmajid Ali Dafallah,Qiaozhen Ma,Eshag Mohamed Ahmed

doi:10.30538/oms2020.0102

Abstract

Random attractors for semilinear reaction-diffusion equation with distribution derivatives and multiplicative noise on \(\mathbb{R}^{n}\)

Highlights

In this paper, we investigate the existence of random attractors for a semilinear reaction-diffusion equation with a nonlinearity having a polynomial growth of arbitrary order p − 1(p ≥ 2), and with distribution derivatives and multiplicative noise defined on unbounded domains
The semilinear reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform a priori estimates for far-field values of solutions as well as the cut-off technique
T In this article, we investigated the existence of the (L2(Rn), L2(Rn))-random attractor and the (L2(Rn), Lp(Rn))-random attractor for the following semilinear reaction-diffusion equation with distribution derivatives and multiplicative noise on Rn: du +dt = ( f (x) − g(u) + Dj f j)dt + bu ◦ dW(t); in R+ × Rn with the initial value condition u(x, 0) = u0(x); x ∈ Rn, (2)

Summary

Introduction

T In this article, we investigated the existence of the (L2(Rn), L2(Rn))-random attractor and the (L2(Rn), Lp(Rn))-random attractor for the following semilinear reaction-diffusion equation with distribution derivatives and multiplicative noise on Rn: du + (λu − ∆u)dt = ( f (x) − g(u) + Dj f j)dt + bu ◦ dW(t); in R+ × Rn (1). In the case of unbounded domains, the existence of random attractors without distribution derivatives was established for the stochastic reaction-diffusion equation with additive noise in [11], and with multiplicative noise in [12]. There are no results on random attractors for stochastic reaction-diffusion equation with distribution derivatives and multiplicative noise on unbounded domain in (L2(Rn), Lp(Rn)). We will use the idea of uniform estimates on the tail of solutions to study the existence of a random attractor of the stochastic reaction-diffusion equation with distribution derivatives and multiplicative noise on unbounded domain. We use · p be the norm of Lp(Rn)(p ≥ 1), |v| the modular of v, m(e), sometimes we write it as |e| the Lebesgue measure of e ⊂ Rn, Rn(|v| ≥ M ) {x ∈ Rn | |v(x)| ≥ M }, and C an arbitrary positive constant, which may be different from line to line and even in the same line, and · , (·, ·) to denote the norm and inner product of L2(Rn), respectively

Preliminaries

Abstract results

The random dynamical system

Existence of random attractors