Global existence and blow-up of solutions to a nonlocal Kirchhoff diffusion problem**This work is supported by NSFC (Grant No. 11201380).

Abstract

In this paper, we revisit the following nonlocal Kirchhoff diffusion problem:∂tu+M([u]s2)LKu=|u|p−2u,inΩ×R+,u(x,t)=0,in(RN\\Ω)×R+,u(x,0)=u0(x),inΩ,where is a bounded domain with Lipschitz boundary, [u]s is the Gagliardo seminorm of u, 0 < s < min{1, N/2}, is a nonlocal integro-differential operator defined in (), which generalizes the fractional Laplace operator (−Δ)s, u0 : Ω → [0, +∞) is the initial function, M : [0, +∞) → [0, +∞) is a continuous function and there exist two constants θ > 1 and m0 > 0 such thatM(σ)⩾m0σθ−1,∀σ∈[0,+∞).This problem has been investigated by Xiang, Rădulescu and Zhang in [], and Ding and Zhou in [] by using potential well method. If , in [], the authors showed the existence of a nontrivial, nonnegative global weak solution, where . However, if , these two papers only studied the model in special cases, the details are as follows: in [], the blow-up conditions for nontrivial, nonnegative weak solution were obtained when J(u0) < 0; in [], the global existence and blow-up conditions for nontrivial, nonnegative weak solution were obtained when J(u0) ⩽ d and M(σ) = m0σθ−1, where J(u0) denotes the initial energy and d > 0 denotes the depth of the potential well (see ()). The main purpose of this paper is to extend the above results to the general case M(σ) ⩾ m0σθ−1, , and the conditions on global existence and finite time blow-up are obtained. Furthermore, the decay estimates for global weak solutions, the growth estimates for blow-up solutions, the upper and lower bounds of blow-up time to blow-up solutions, the behavior of the energy functional as t → T (where T denotes the blow-up time) are studied. Moreover, some blow-up conditions independent of d and some equivalent conditions for the weak solutions existing globally or blowing up in finite time are investigated. Finally, the global existence and finite time blow-up results with high initial energy (i.e., J(u0) > d) are obtained.