Global existence and blow-up of solutions to a class of nonlocal parabolic equations

This paper is devoted to study a class of nonlocal parabolic equations, which was considered in Liu and Ma (2014) and Li and Liu (2017), where the case of initial energy J(u(0)) <= d (d is the mountain pass level) was discussed, the conditions on global existence or blow-up, the vacuum region and the asymptotic behavior of the solutions were studied. We extend their results on the following three aspects: Firstly, we explicitly give the vacuum region, the global existence region and the blow-up region when J(u(0)) < d, that is, there exist three regions <(U)over tilde>(e), G(e) and B-e such that H-0(1)(Omega) = (U) over tilde (e) boolean OR G(e) boolean OR B-e, and 1. (U) over tilde (e) is a vacuum region, i.e., the solution does not belong to (U) over tilde (e); 2. G(e) is an invariant region, the solution exists globally and decays to 0 exponentially if the initial value belongs to G(e); 3. B-e is an invariant region, the solution blows up in finite time if the initial value belongs to B-e. Secondly, we estimate the upper and lower bounds of the blow-up time and blow-up rate for the blow-up solutions when J(u(0)) <= d. Thirdly, we consider the asymptotic behavior of the solutions when J(u(0)) > d. By constructing two sets psi(alpha), and phi(alpha), we prove that the solution blows up in finite time if the initial value belongs to phi(alpha), while the solution exists globally and tends to zero as time t -> +infinity when the initial value belongs to phi(alpha). (C) 2019 Elsevier Ltd. All rights reserved.