Compact invariant sets for some quasilinear nonlocal Kirchhoff strings on ℝ N

Abstract

We consider the quasilinear nonlocal dissipative Kirchhoff String problem with the initial conditions u(x, 0) = u 0(x) and u t (x, 0) = u 1(x), in the case where N ≥ 3, δ ≥ 0, f(u) = |u| a u for example, and (φ(x))−1 ∈ L N/2(ℝ N ) ∩ L ∞(ℝ N ) is a positive function. The purpose of our work is to study the long-time behaviour of the solution of this equation. The compactness of the embeddings , is widely applied.