Pullback attractors for a class of second-order delay evolution equations with dispersive and dissipative terms on unbounded domain

Abstract

In this article, we investigate the long-time behavior for the ill-posed problems $$\frac{{\partial }^{2}u}{\partial t^2}+\frac{\partial u}{\partial t}+\lambda u-\mathrm{\Delta }u-\mathrm{\Delta }\frac{\partial u}{\partial t}-\mathrm{\Delta }\frac{{\partial }^{2}u}{\partial t^2}=f\left(t,u\left(x,t-\rho \left(t\right)\right)\right)+g\left(t,x\right),\mathrm{in}\left(\tau ,+\infty \right)\times {\mathbb{R}}^N,$$ with some hereditary characteristics. First, we establish the existence of solutions for the second-order non-autonomous evolution equation by the standard Faedo-Galerkin methods, but without any Lipschitz conditions on the nonlinear term $f\left(\cdot \right)$. Then, by proving the $\mathfrak{D}$-pullback asymptotically upper-semicompact property for the multivalued process $\left\{U\left(t,\tau \right)\right\}$, we establish the existence of pullback attractors ${\mathcal{A}}_{C_{H^1\left({\mathbb{R}}^N\right),H^1\left({\mathbb{R}}^N\right)}}$ in the Banach spaces $C_{H^1\left({\mathbb{R}}^N\right),H^1\left({\mathbb{R}}^N\right)}$ for the multi-valued process generated by a class of second-order non-autonomous evolution equations with delays and ill-posedness.