In this paper, we consider the following Kirchhoff-type diffusion problem involving the fractional Laplacian and logarithmic nonlinearity at high initial energy level: ut+[u]s2(θ−1)(−Δ)su=|u|q−2uln|u|(x,t)∈Ω×R+,u(x,t)=0(x,t)∈(RN∖Ω)×R+,u(x,0)=u0(x)x∈Ω, where (−Δ)s is the fractional Laplacian with s∈(0,1),N>2s, [u]s is the Gagliardo seminorm of u, Ω⊂RN is a bounded domain with Lipschitz boundary, 1⩽θ<N/(N−2s), 2θ<q<2s∗. Based on the potential well theory, a sufficient condition is given for the existence of global solutions that vanish at infinity or solutions that blow up in finite time under some appropriate assumptions. In particular, the existence of ground state solutions for the above stationary problem is obtained by restricting the related discussion on Nehari manifold.

中文翻译：

---

具有对数非线性的简并Kirchhoff型分数扩散问题

在本文中，我们考虑了以下在高初始能级下涉及分数拉普拉斯和对数非线性的基尔霍夫型扩散问题：ut + [u] s2（θ-1）（-Δ）su = | u | q-2uln | u |（x，t）∈Ω×R +，u（x，t）= 0（x，t）∈（RN∖Ω）×R +，u（x，0）= u0（x）x∈Ω，其中（ -Δ）s是s∈（0,1），N> 2s的分数拉普拉斯算子，[u] s是u的Gagliardo半范数，Ω⊂RN是Lipschitz边界的有界域，1⩽θ<N /（ N-2s），2θ<q <2s ∗。基于势阱理论，给出了存在于无穷大的整体解或在某些适当假设下在有限时间内爆炸的整体解的充分条件。尤其是，通过限制有关Nehari流形的相关讨论，可以获得上述平稳问题的基态解的存在。