Long-time behavior of solutions for a fractional diffusion problem

Abstract

Abstract This paper deals with the asymptotic behavior of solutions to the initial-boundary value problem of the following fractional p-Kirchhoff equation: u t + M ( [ u ] s , p p ) ( − Δ ) p s u + f ( x , u ) = g ( x ) in Ω × ( 0 , ∞ ) , $$ u_{t}+M\bigl([u]_{s,p}^{p}\bigr) (-\Delta )_{p}^{s}u+f(x,u)=g(x)\quad \text{in } \Omega \times (0, \infty ), $$ where Ω ⊂ R N $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with Lipschitz boundary, N > p s $N>ps$ , 0 < s < 1 < p $0< s<1<p$ , M : [ 0 , ∞ ) → [ 0 , ∞ ) $M:[0,\infty )\rightarrow [0,\infty )$ is a nondecreasing continuous function, [ u ] s , p $[u]_{s,p}$ is the Gagliardo seminorm of u, f : Ω × R → R $f:\Omega \times \mathbb{R}\rightarrow \mathbb{R}$ and g ∈ L 2 ( Ω ) $g\in L^{2}(\Omega )$ . With general assumptions on f and g, we prove the existence of global attractors in proper spaces. Then, we show that the fractal dimensional of global attractors is infinite provided some conditions are satisfied.

Keywords

• Fractional p-Laplacian
• Global attractors
• Fractal dimension