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.

Keywords:

• Quasilinear hyperbolic equations
• Kirchhoff strings
• Global attractor
• Unbounded domains
• Generalised sobolev spaces
• Weighted L ^p spaces

AMS Subject Classifications::

• 35A07
• 35B30
• 35B40
• 35B45
• 35L15
• 35L70
• 35L80
• 47F05
• 47H20

Acknowledgements

We would like to thank Professor Hiroshi Matano, for his valuable comments. This work was partially financially supported by a grant from Papakyriakopoulos Legacy in the Department of Mathematics at NTU, Athens and by the Thalis project No. 65/1211 from the Committee of Fundamental Research at NTUA, Athens, Greece.