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 {ut + M([u](s,p)(p))(-Delta)(p)(s)u = lambda vertical bar u vertical bar(r-2)u-mu vertical bar u vertical bar(q-2)u (x, t) is an element of Omega x (0, infinity), u = 0 (x,t) is an element of (R-N \ Omega) x (0, infinity), u(x, 0) = uo(x) x is an element of Omega, where [u](s,p) is the Gagliardo seminorm of u, Omega subset of R-N is a bounded domain with Lipschitz boundary, (-Delta)(p)(s) is the fractional p-Laplacian with 0 < s < 1 < p < 2, M : [0, infinity) -> (0, infinity) is a continuous function, 1 < q <= 2, r > 1 and lambda, mu > 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. (C) 2019 Elsevier Inc. All rights reserved.