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 ∂2u∂t2+∂u∂t+λu−Δu−Δ∂u∂t−Δ∂2u∂t2=f(t,u(x,t−ρ(t)))+g(t,x),in(τ,+∞)×RN,\frac{{\partial }^{2}u}{\partial {t}^{2}}+\frac{\partial u}{\partial t}+\lambda u-\Delta u-\Delta \frac{\partial u}{\partial t}-\Delta \frac{{\partial }^{2}u}{\partial {t}^{2}}=f\left(t,u\left(x,t-\rho \left(t)))+g\left(t,x),\hspace{1.0em}{\rm{in}}\hspace{0.33em}\left(\tau ,+\infty )\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(⋅)f\left(\cdot ). Then, by proving the D{\mathfrak{D}}-pullback asymptotically upper-semicompact property for the multivalued process {U(t,τ)}\left\{U\left(t,\tau )\right\}, we establish the existence of pullback attractors ACH1(RN),H1(RN){{\mathcal{A}}}_{{C}_{{H}^{1}\left({{\mathbb{R}}}^{N}),{H}^{1}\left({{\mathbb{R}}}^{N})}} in the Banach spaces CH1(RN),H1(RN){C}_{{H}^{1}\left({{\mathbb{R}}}^{N}),{H}^{1}\left({{\mathbb{R}}}^{N})} for the multi-valued process generated by a class of second-order non-autonomous evolution equations with delays and ill-posedness.

Keywords

• non-autonomous semi-linear second-order evolution
• pullback attractors
• variable delays
• multi-valued process
• 35b41
• 35q35
• 35b40
• 37l05