Non-autonomous nonlocal partial differential equations with delay and memory

Abstract

The paper addresses a kind of non-autonomous nonlocal parabolic equations when the external force contains hereditary characteristics involving bounded and unbounded delays. First, well-posedness of the problem is analyzed by the Galerkin method and energy estimations in the phase space C ρ (X). Moreover, some results related to strong solutions are proved under suitable assumptions. The existence of stationary solutions is then established by a corollary of the Brower fixed point theorem. By constructing appropriate Lyapunov functionals in terms of the characteristic delay terms, a deep analysis on stability and attractiveness of the stationary solutions is established. Furthermore, the existence of pullback attractors in L 2 (Ω), with bounded and unbounded delays, is shown. We emphasize that, to prove the existence of pullback attractors in the unbounded delay case, a new phase space, E γ , has to be constructed.

Asymptotic behavior of nonlocal partial differential equations with long time memory

“…Obviously, these conditions are more restrictive than the previous (i)-(iii).We also want to emphasize that if condition (11) were adopted in [17] instead of (2.3)-(2.4), the method to prove the main results of [17] could not be used successfully.…”

“…Inspired by the above work, the dynamics of the following non-autonomous nonlocal partial differential equations with delay and memory was investigated in [17] by using the Galerkin method and energy estimations,…”

“…According to standard existence theory for ODE, there exists a continuous solution of ( 17)-( 18) on some interval (τ, T n ). Moreover, a priori estimation implies T n = ∞, for more details, see [17].…”

Exponential decay for nonlinear abstract evolution equations with a countably infinite number of time‐dependent time delays

“…In particular, we refer to Alabau-Boussouira 22 for a Timoshenko model for beams in one dimension. Finally, we refer also to Caraballo and Han, Liu et al, Marin-Rubio et al, and Xu et al [23][24][25][26] for other applications to delayed systems.…”

Uniform attractors for nonautonomous reaction-diffusion equations with the nonlinearity in a larger symbol space

“…For nonautonomous PDEs, the situation becomes more complicated since the explicit dependence of the system on the initial time s and the final time t. As far as we know, mainly two kinds of methods can be used to generalize the concepts of attractors to the nonautonomous PDEs. One of them describes the nonautonomous attractor A as a time-dependent set A = A(t), t ∈ R. This leads to the concept of pullback attractors, see [2], [6], [16], [18]. And it can also be well adapted to study the Random/Stochastic PDEs, see, e.g., [4].…”