Strong global attractor for a quasilinear nonlocal wave equation on ℝN

Abstract

We study the long time behavior of solutions to the nonlocal quasilinear dissipative wave equation utt - ϕ(x) ∥∇u(t)∥2 ∆u + δut + |u|α u = 0, in ℝN, t ≥ 0, with initial conditions u(x, 0) = u0(x) and ut(x, 0) = u1(x). We consider the case N ≥ 3, δ > 0, and (ϕ(x))-1 a positive function in LN/2(ℝN) ∩ L∞(ℝN). The existence of a global attractor is proved in the strong topology of the space D1,2(ℝN) x L2g(ℝN).