Local existence, global existence and blow-up of solutions to a nonlocal Kirchhoff diffusion problem*
Hang Ding, Jun Zhou
Abstract
Abstract 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 m 0 > 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 ( u 0 ) < 0, where J ( u 0 ) 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).