Uniform attractors for multi-valued process generated by non-autonomous nonclassical diffusion equations with delay in unbounded domain without uniqueness of solutions

Uniform attractors of nonclassical diffusion equations on ℝN with memory and singularly oscillating external forces

“…In the last two decades, many researchers have concentrated on the theory of attractors for dynamical systems. The existence and long-time behavior of solutions to nonclassical diffusion equations have been investigated extensively in various cases, such as in the cases of autonomous (see other works [5][6][7][8][9][10][11] ) and of nonautonomous (see other studies 7,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], and even in the case with finite delay (see other works [21][22][23] ). Besides, these equations with singularly oscillating external force have been also studied in Anh and Toan, 24 where they obtained the boundedness and the upper semicontinuity of uniform attractors in the case when the domain is unbounded and the nonlinearity is of Sobolev type.…”

“…21 ( )e − (t− ) + C( 1 + m 2 + M 2 0 ).Recalling that z(t) = (v(t), t 1 (s)) + (w(t), t 2(s)) and using (4.20) once again, we have for all ||z|| e − (t− ) ||z || C(1 + m 2 + M 2 0 ), ∀t ≥ .…”

Dynamics of fractional nonclassical diffusion equations with delay driven by additive noise on $ \mathbb{R}^n $

“…When α = 1, the fractional Laplacian (−∆) α becomes the standard Laplace operator −∆. In this special case, the deterministic attractors of the equation have been investigated in [4,5,7,18,19,54,57,63,69,70,73], and the random attractors have been studied in [8,27,41,76] and [65,66] for white noise and colored noise, respectively. Recently, the existence of random attractors of the stochastic nonclassical diffusion equations with time-varying delay has been reported in [22].…”

“…In this case, the response of a system depends not only on its current state, but also on the past history of the system, see, e.g., [35,51,52,72]. It is worth mentioning that the attractors of differential equations with delay have been investigated in [7,20,18,19,37,39,64] in the deterministic case and in [8,15,17,16,27,31,42,67,68,71,74,75] in the stochastic case.…”