Trajectory Attractors for Reaction-diffusion Systems
نویسندگان
چکیده
(1) ∂tu = a∆u− f0(u, t) + g0(x, t), u|∂Ω = 0 (or ∂u/∂ν|∂Ω = 0) where u = u(x, t) = (u, . . . , u ), x ∈ Ω b R, t ≥ 0, f0(v, s) = (f 0 , . . . , f 0 ), (v, s) ∈ R × R+, g0(x, s) = (g 0 , . . . , g 0 ), x ∈ Ω, s ≥ 0. We assume that the matrix a and the functions f0, g0 satisfy some general conditions (see Section 2). These conditions provide the existence of a solution u of the Cauchy problem for the system (1) (u|t=0 = u0, u0 ∈ H = (L2(Ω)) ). However, this solution can be non-unique because we do not suppose any Lipschitz conditions for f0(v, s) with respect to v. The pair of functions (f0(v, s), g0(x, s)) = σ0(s) is called the symbol of equation (1). To construct a trajectory attractor for (1), we consider the family of
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