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Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains. (English) Zbl 1173.37065

The paper is concerned with the stochastic FitzHugh-Nagumo system defined on \(\mathbb{R}^n\) \[ \begin{cases} du+(\lambda u-\Delta u+\alpha v)dt=(f(x,u)+g)dt+\phi_1dw_1,\,\,\,x\in \mathbb{R}^n,\,\,\,t>0,\\ dv+(\delta v-\beta u)dt=hdt+\phi_2dw_2,\,\,\,x\in \mathbb{R}^n,\,\,\,t>0,\end{cases}.(S) \] where \(\lambda,\,\alpha,\,\delta\) and \(\beta\) are positive constants, \(g\in L^2(\mathbb{R}^n)\), \(h\in H^1(\mathbb{R}^n)\), \(\phi_1\in H^2(\mathbb{R}^n)\cap W^{2,p}(\mathbb{R}^n)\) for some \(p\geq 2\), \(\phi_2\in H^1(\mathbb{R}^n)\), \(f\) is a nonlinear function that satisfies some dissipative conditions, \(w_1\) and \(w_2\) are independent two-sided real-valued Wiener processes on a probability space. The author defines a continuous random dynamical system for system \((S)\) with the initial conditions \(u(x,0)=u_0(x),\,\,v(x,0)=v_0(x)\), for \(x\in \mathbb{R}^n\). Then he derives uniform estimates on the solutions of \((S)\) when \(t\to\infty\), that are necessary to prove the existence of bounded absorbing sets and the asymptotic compactness of the random dynamical system. Finally the author proves the existence of a pullback random attractor for the random dynamical system associated to \((S)\).

MSC:

37L55 Infinite-dimensional random dynamical systems; stochastic equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
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