zbMATH — the first resource for mathematics

Non-autonomous attractors for integro-differential evolution equations. (English) Zbl 1185.45016
In the first part of this paper, the authors provide sufficient conditions for the existence of pullback attractors for so-called multivalued non-autonomous dynamical systems (MNDS). The pullback attractors are defined here with respect to a universe of subsets of the state space with sub-exponential growth. Next they consider the following evolution equation
\[ \frac{dy}{dt}=Ay+f(t,y_t), \]
where \(A\) is the generator of a \(C_0\) contraction semigroup \((e^{At})_{t\geq 0}\) on a separable Banach space \((H,\|\cdot\|)\) such that
\[ \|e^{At}x\|\leq\|x\|e^{-\alpha t},\;\;\text{for some } \alpha>0\;\;\text{and every } t\geq 0, \]
the operators \(e^{At}\) are compact for \(t>0\) and \(y_t:(-\infty,0]\to H\) is defined as \(y_t(s)=y(t+s)\), \(s\in (-\infty,0]\).
It is proved that under suitable assumptions on \(f\) the initial value problem \((IVP)_{t_0,\varphi}\) for the above equation possesses a mild solution in the function space \(C_{\gamma}=\{u\in C((-\infty,0];H):\;\lim_{\tau\to -\infty}u(\tau) e^{\gamma\tau}\,\text{exists}\}\), where \(\gamma>\alpha\).
Moreover, it is proved that under suitable assumptions an MNDS generated by the above equation has a pullback \(\mathcal D\)-attractor \(A\) in the suitable set \(C(C_{\gamma})\).
As examples the authors consider an ordinary integro-differential equation and an integro-differential reaction-diffusion equation.

45N05 Abstract integral equations, integral equations in abstract spaces
34K25 Asymptotic theory of functional-differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
45J05 Integro-ordinary differential equations
45K05 Integro-partial differential equations
34G20 Nonlinear differential equations in abstract spaces
45G10 Other nonlinear integral equations
Full Text: DOI