Nonlocal Kirchhoff problems: Extinction and non-extinction of solutions.

In this paper, we discuss the extinction and non-extinction properties of solutions for the following fractional p -Kirchhoff problem { u t + M ([ u ] s , p p) (− Δ) p s u = λ | u | r − 2 u − μ | u | q − 2 u (x , t) ∈ Ω × (0 , ∞) , u = 0 (x , t) ∈ (R N ∖ Ω) × (0 , ∞) , u (x , 0) = u 0 (x) x ∈ Ω , where [ u ] s , p is the Gagliardo seminorm of u , Ω ⊂ R N is a bounded domain with Lipschitz boundary, (− Δ) p s is the fractional p -Laplacian with 0 < s < 1 < p < 2 , M : [ 0 , ∞) → (0 , ∞) is a continuous function, 1 < q ≤ 2 , r > 1 and λ , μ > 0. Under suitable assumptions, we obtain the extinction of solutions. To get more precisely decay estimates of solutions, we develop the Gagliardo-Nirenberg inequality. Moreover, the non-extinction property of solutions is also investigated. [ABSTRACT FROM AUTHOR]