Attractors and their continuity for an extensible beam equation with rotational inertia and nonlocal energy damping

Abstract

This paper investigates the well-posedness of global solutions, the existence of global and exponential attractors and their continuity for an extensible beam equation with rotational inertia and nonlocal energy damping in Ω⊂RN: (1−αΔ)utt+Δ2u−ϕ(‖∇u‖2)Δu−M(‖Δu‖2+‖ut‖2)Δut+f(u)=h, where α∈[0,1] is a rotational coefficient. We find a better subcritical index p⁎:=N+2N−4, with N≥5, and show that when the growth exponent p of the nonlinearity f(u) is up to the range: 1≤p≤p⁎, the equation is well-posed. In particular, when 1≤p<p⁎: (i) the related solution semigroup Sα(t) has a global attractor Aα for each α∈[0,1] and the family of {Aα}α∈[0,1] is continuous on α in a residual subset of [0,1], and upper semi-continuous on [0,1]; (ii) Sα(t) has an exponential attractor Aexpα for each α and the family of {Aexpα}α∈[0,1] is Hölder continuous on α. The main contributions of this paper are that: (a) it breaks through the existing growth restriction: 1≤p≤NN−4 on this issue in literature; (b) it gives a new criterion on the continuity (rather than only upper semi-continuity) of the global attractors in a residual subset of the parameter space and its application; (c) it provides a new method to establish the continuity of the exponential attractors for pure hyperbolic model.