Random attractor for the reaction-diffusion equation perturbed by a stochastic càdlàg process. (Ukrainian, English) Zbl 1114.37028

Kapustyan, O. V.; Valero, J.; Pereguda, O. V.

Teor. Jmovirn. Mat. Stat. 73, 52-62 (2005); translation in Theory Probab. Math. Stat., Vol 73, 57-69 (2006).

The authors consider the problem
\[ \begin{cases} \frac{ \partial{u(t,x)}}{\partial{t}}= a\Delta u(t,x)-f(u(t,x))+h(x)+g(u(t,x))\xi(t,w),\\ u|_{\partial{Q}}=0,\quad u|_{t=0}=u_0(x), \end{cases} \]
where \(a>0\), \(Q\subset \mathbb R^n\) is a bounded set with smooth boundary, \(f,g\in C(\mathbb R)\), \(h\in L_2(Q)\), \(\xi(t,w)\) is a stochastic càdlàg process. They study this stochastically perturbed reaction-diffusion equation by using methods of the theory of stochastic attractors. They show that solutions of the equation form a multivalued random dynamical system. The existence and uniqueness of a random attractor is proved. For the related results see H. Crauel [Ann. Mat. Pura Appl., IV. Ser. 176, 57–72 (1999; Zbl 0954.37027)], K. R. Schenk-Hoppé [Discrete Contin. Dyn. Syst. 4, No. 1, 99–130 (1998; Zbl 0954.37026)], T. Caraballo, J. A. Langa and J. Valero [Nonlinear Anal., Theory Methods Appl. 48, No. 6(A), 805–829 (2002); addendum ibid. 61, No. 1–2(A), 277–279 (2005; Zbl 1004.37035)].

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

37H10	Generation, random and stochastic difference and differential equations
34F05	Ordinary differential equations and systems with randomness
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
37L30	Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems

Keywords:

stochastically perturbed reaction-diffusion equation; stochastic attractors; multi-valued random dynamic system

Citations:

Zbl 0954.37027; Zbl 0954.37026; Zbl 1004.37035