The interplay between fractional damping and nonlinear memory for the plate equation

In this paper, we study the interplay between a fractional damping (- Delta)(theta)u(t) with theta is an element of [0,1/2), and a nonlinear memory term applied to a plate equation: u(tt) - Delta u(tt) - Delta u + Delta(2)u + (-Delta)(theta)u(t) = integral(t)(0) (t-s)(-gamma)vertical bar u(S, .)vertical bar(p) ds, t > 0, x is an element of R-n, in space dimension n = 1, 2, 3, 4. In different scenarios, we prove the global existence of small data solutions in C([0, infinity), H-2) boolean AND C-1 ([0, infinity), H-1) for supercritical powers, where the critical exponent is determined by the fractional power of the damping term and by the order of the fractional integration in the memory term. In one case, we also feel the influence of the regularity-loss type decay, which is due to the presence of the rotational inertia term -Delta u(tt) in the plate equation.