Local existence, global existence and blow-up of solutions to a nonlocal Kirchhoff diffusion problem^*

In this paper, we study the following diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator where [u]_s is the Gagliardo seminorm of u, is a bounded domain with Lipschitz boundary, , is a nonlocal integro-differential operator defined in (1.2), which generalizes the fractional Laplace operator , is the initial function, and is a continuous function and there exist two constants m₀  >  0 and such that

As is well-known, the nonlocal Kirchhoff problem was first introduced and motivated in Fiscella and Valdinoci (2014 Nonlinear Anal. 94 156–70) and the above problem was studied by Xiang et al (2018 Nonlinearity 31 3228–50), the main results of Xiang et al (2018 Nonlinearity 31 3228–50) are as follows:

 The local existence of nontrivial, nonnegative weak solution for , where .

 The blow-up conditions for nontrivial, nonnegative weak solution when J(u₀)  <  0, where J(u₀) denotes the initial energy.

The main purpose of this paper is to extend the above results and we get:

 The global existence of nontrivial, nonnegative weak solution for any .

 The global existence and blow-up conditions for nontrivial, nonnegative weak solution when for the case , where d is a positive constant given in (1.13).