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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mo>∂</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:msup> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mo>∂</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:mi>t</m:mi> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:mi>λ</m:mi> <m:mi>u</m:mi> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mfrac> <m:mrow> <m:mo>∂</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:mi>t</m:mi> </m:mrow> </m:mfrac> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mo>∂</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:msup> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo>−</m:mo> <m:mi>ρ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="1.0em"/> <m:mi mathvariant="normal">in</m:mi> <m:mspace width="0.33em"/> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>τ</m:mi> <m:mo>,</m:mo> <m:mo>+</m:mo> <m:mi>∞</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>×</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>⋅</m:mo> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> f\left(\cdot ) . Then, by proving the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="fraktur">D</m:mi> </m:math> {\mathfrak{D}} -pullback asymptotically upper-semicompact property for the multivalued process <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mi>U</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>τ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> \left\{U\left(t,\tau )\right\}