The paper investigates the well-posedness and the complete regularity of the weak solutions, and the longtime dynamics for the structurally damped wave equation with almost-linear h(x, ut) and supercritical nonlinearity g(x, u) on ℝN (N ⩾ 3): utt − Δu + (− Δ)αut + h(x, ut) + g(x, u) = f, where the perturbed parameter α ∈ (1/2, 1) is a dissipative index determining the dissipative strength. We show that when the growth order p of the nonlinearity g(x, u) is up to the supercritical range: \(p^{\ast}<p<\bar{p}_{\alpha}\) , (i) the Cauchy problem of the model is well-posed and its weak solution is exactly the strong one; (ii) the related dynamical system (Sα(t), \(\cal{H}\) ) possesses a strong ( \({\cal{H}},{\cal{H}}_{2\alpha}\) )-global attractor \({\cal{A}}_{\alpha}\) , which is also the standard global attractor of optimal regularity for each α ∈ (1/2, 1), where \({\cal{H}}:=(H^{1}(\mathbb{R}^{N})\cap L^{p+1}(\mathbb{R}^{N}))\times L^{2}(\mathbb{R}^{N})\) is the energy space while \({\cal{H}}_{2\alpha}\) is the strong solution space; (iii) the family of strong ( \({\cal{H}}, {\cal{H}}_{2\alpha}\) )-global attractors \({\cal{A}}_{\alpha}\) is upper semicontinuous at each point α0 ∈ (1/2, 1) in \({\cal{H}}_{2\alpha_{0}}\) -topology. The method developed here provides a passage to deal with the supercritical nonlinearity to get the above-mentioned results which extend those on this issue in the previous literature.
