Global existence and finite time blow-up for a class of fractional p-Laplacian Kirchhoff type equations with logarithmic nonlinearity

Abstract: In this paper, we study the initial-boundary value problem for a class of fractional pLaplacian Kirchhoff diffusion equation with logarithmic nonlinearity. For both subcritical and critical states, by means of the Galerkin approximations , the potential well theory and the Nehari manifold, we prove the global existence and finite time blow-up of the weak solutions. Further, we give the growth rate of the weak solutions and study ground-state solution of the corresponding steady-state problem.


Introduction
In this paper, we study the global existence and blow-up for the following nonlinear fractional p-Laplacian Kirchhoff diffusion equation.
in Ω, u = 0, on ∂Ω × (0, T ), where s ∈ (0, 1), 1 < p < N/s (N is space latitude and satisfies N ≥ 1), 1 ≤ α < p * s /p, p * s = N p N−sp , Ω ⊂ R N is a bounded domain with Lipschitz boundary, [u] s,p is Gagliardo seminorm of u, (−∆) s p is the fractional p-Laplacian operator and satisfies |u(x) − u(y)| p−2 (u(x) − u(y)) |x − y| N+sp dy, where u(x) ∈ C ∞ and u(x) has compact support in Ω, B β (x) ⊂ R N is the set of spheres with x as the center and β as the radius. Let M(t) = a + bt α−1 (t ≥ 1), where a and b are constants and satisfy a ≥ 0, b > 0, in this paper we let a = 0, b = 1, t = [u] p s,p , i.e. M([u] p s,p ) = [u] αp−p s,p . Let Ω ⊂ R N be a bounded domain with Lipschitz boundary, and use · r to denote the norm of Lebesgue space L r (Ω), where r ∈ (0, +∞). In view of the idea from [1], we define linear space W s,p (Ω) as a fractional Sobolev space and discuss in the fractional Sobolev space W s,p 0 (Ω), W s,p 0 (Ω) = {u ∈ W s,p (Ω) | u(t) = 0 a.e.in R N \Ω}.
The norm of space W s,p (Ω) is defined as the following equation |u(x) − u(y)| p |x − y| N+sp dxdy 1 p .
According to the research result from [2], it can be concluded that |u(x) − u(y)| p |x − y| N+sp dxdy 1 p .
Until now, many scholars have studied the fractional Laplacian problem. In [1], the authors studied the fractional Kirchhoff-type problem.
x ∈ Ω, where s ∈ (0, 1), q ∈ (2θ, 2 * ) , θ ∈ [1, N/(N − 2s) , N > 2s, Ω ⊂ R N is a bounded domain with Lipschitz boundary. They use the variational principle, Nehari manifold and potential well method to study the finite time blow-up of the solution of the problem (1.2) and the sufficient conditions for the global existence.
In [2], the authors studied the following initial value problem: . By the method of the mountain pass lemma and Nehari manifold, the existence of the minimum energy solutions are obtained. For the steady-state equation, it is worth emphasizing that for the local minimum solution, Liu and Liao and Pan used a brand-new method in [29], and also obtained some properties of the corresponding equation.
In [3], the authors discuss the following equations where L K is a nonlocal integro-differential operator. They used the Galerkin approximation method and the potential well to prove the existence of a global weak solution with subcritical and critical states. According to the differential inequality, the blow-up solution of the equation is given. At the same time, the lower bound of the solution of problem (1.4) and the existence of the ground state solution of the corresponding steady-state problem was discussed. More details for fractional Laplacian equations with logarithmic nonlinearity can refer to [30][31][32]. Inspired by the above references, we study problem (1.1). Compared with problem (1.2), we consider the case of 1 < p < N/s and discuss the global existence and finite time blow-up of the solutions. If p = 2 and 1 < α < N/(N − 2s), we can turn problem (1.1) into problem (1.2). Compared with problem (1.3) and (1.4), we study the global existence and finite time blow-up of the solutions for fractional p-Laplacian Kirchhoff type equation with logarthmic nonlinearity. By the methods of the variational principle and Nehari manifold, as well as combining with the relevant theories and properties of the fractional Sobolev space definition, we consider both E(u 0 ) < h and E(u 0 ) = h cases and prove the global existence and the finite time blow-up of the solution for problem (1.1), and discuss the growth rate of the weak solutions and study ground-state solution of the corresponding steady-state problem.
The rest of the paper is organized as follws. In Section 2, we give some related definitions and lemmas. In Section 3, we prove the global existence and the finite time blow-up of the solution with subcritical state E(u 0 ) < h of the problem (1.1), at the same time, we give the growth rate of the solutions and study ground-state solution of the corresponding steady-state problem. In Section 4, we prove the global existence and the finite time blow-up of the solution with critical state E(u 0 ) = h of the problem (1.1).

Preliminaries and Lemmas
In this section, we give some related definitions and Lemmas needed to prove the conclusions later. First of all, we define where D * is the best embedding constant for W s,p 0 (Ω) → L αp+ (Ω), ∈ (0, p * s − αp). The energy functional is The Nehari functional is And then, we define some sets as follows:
we can conclude that (i) holds.
(iv) According to the (2.11) and Proof. Set g(t) = ln t − t e , t ∈ [1, +∞). By a simple derivative calculation of the function, we get Obviously t * is the maximum point of function g(t), thus g(t) ≤ g( t * ) = 0 for all t ∈ [1, +∞). This proves the above inequality.
Proof. By Lemma 2.2 and the definition of Nehari function I(u), we have we can get If s ∈ (0, 1),1 < p < N/s holds, then the functionals E(u) and I(u) are well-defined and continuous on W s,p 0 (Ω). Moreover, where 1 ≤ α < p * s /p, then we can claim that E(u) and I(u) are well-defined in W s,p 0 (Ω). Further, Similar to Lemma 2.3 in [2], one can prove the that E(u) ∈ C 1 (W s,p 0 (Ω), R), and I(u) = E (u), u for all u ∈ W s,p 0 (Ω). Lemma 2.5 [33] Let s ∈ (0, 1) and p ∈ [1, +∞) such that sp < N. Let q ∈ [1, p * ), Ω ⊆ R N be a bounded extension domain for W s,p (Ω) and ζ be a bounded subset of L p (Ω). Suppose that , and assume that u(x, t) is the solution of problem (1.1), then d dt u 2 2 = −2I(u). Proof. By the definition of the weak solution, we have Thus, we obtain d dt u 2 2 = −2I(u).
(Ω) and u t ∈ L 2 0, T ; L 2 (Ω) , for all v ∈ W s,p 0 (Ω), t ∈ (0, T ), the following equation holds Definition 2.2 (Maximal existence time) Let u(t) be a solution of problem (1.1), we define the maximal existence time T of u(t) as follows:

Subcritical state (E(u 0 ) < h)
In this section, we use Galerkin approximation, potential well theory, Nehari manifold to study the problem (1.1) the global existence of the solution , blow-up in finite time of the solution with subcritical state E(u 0 ) < h of the problem (1.1). In addition, we discuss the growth rate of the weak solution. Finally, we also discuss the ground state solution of the corresponding steady-state problem.
According to the above discussion, there exists a subsequence of {u n }. For the convenience, we still use {u n } to represent the subsequence, and further obtain u n v weakly in W s,p 0 (Ω), u n v strongly in L αp (Ω), u n → v a.e. in Ω.
By Sobolev embedding theorem and logarithmic inequality, we have Ω * |u n | αp ln |u n |dx = − Ω * (|u n |<1) |u n | αp ln |u n |dx + Ω * (|u n |≥1) |u n | αp ln |u n |dx where Ω * ⊂ Ω is a measurable subset, thus we can get that |u n | αp ln |u n | is uniformly bound. By integral convergence theorem and |u n | αp ln |u n | → |v| αp ln |v| a.e. in Ω , we have In other words 0 < Ω |v| αp ln |v|dx, thus, we can conclude that v 0. According to (ii) of Lemma 2.1, there exists a ρ * > 0 satifying ρ * v ∈ N, I(ρ * v) = 0, i.e. By simply calculating, we can get This contradicts our definition of E(ρ * v) ≥ h , so we can conclude Next, we prove v is the ground state solution of the steady-state problem. From the above proof, we obtain E(v) = h, I(v) = 0. By the theory of Lagrange multipliers, there exists ξ ∈ R such that Since We get that v is the ground state solution of the steady-state problem.

Conclusions
In this work, we study the initial-boundary value problem for a class of fractional p-Laplace Kirchhoff diffusion equation with logarithmic nonlinearity. For both subcritical and critical states, by means of the Galerkin approximations , the potential well theory and the Nehari manifold, we prove the global existence and finite time blow-up of the weak solutions. Further, we give the growth rate of the weak solutions and study ground-state solution of the corresponding steady-state problem. Compared with problem (1.2), we consider the case of 1 < p < N/s and discuss the global existence and finite time blow-up of the solutions. Compared with problem (1.3) and (1.4), we study the well-posedness of the solution for the fractional p-Laplacian Kirchhoff type evolution equation with logarithmic nonlinearity.