LAUSR

Abstract 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.