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

1.   Introduction

In this paper, we study the global existence and blow-up for the following nonlinear fractional $p$-Laplacian Kirchhoff diffusion equation.

where $s\in(0, 1)$, $1 < p < N/s$ ($N$ is space latitude and satisfies $N \geq 1$), $1\leq\alpha < p^*_s/p$, $p^*_s = \frac{Np}{N-sp}$, $\Omega\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $[u]_{s, p}$ is Gagliardo seminorm of $u$, $(-\Delta)^s_p$ is the fractional $p$-Laplacian operator and satisfies

where $u(x)\in C^{\infty}$ and $u(x)$ has compact support in $\Omega$, $B_\beta (x)\subset \mathbb{R}^N$ is the set of spheres with $x$ as the center and $\beta$ as the radius. Let $M(t) = a+bt^{\alpha-1}(t\geq 1)$, where $a$ and $b$ are constants and satisfy $a\geq0$, $b > 0$, in this paper we let $a = 0$, $b = 1$, $t = [u]^p_{s, p}$, i.e. $M([u]^p_{s, p}) = [u]^{\alpha p-p}_{s, p}$. Let $\Omega\subset \mathbb{R}^N$ be a bounded domain with Lipschitz boundary, and use $\|\cdot\|_r$ to denote the norm of Lebesgue space $L^r(\Omega)$, where $r\in(0, +\infty)$. In view of the idea from ^[1], we define linear space $W^{s, p}(\Omega)$ as a fractional Sobolev space and discuss in the fractional Sobolev space $W^{s, p}_0(\Omega)$,

The norm of space $W^{s, p}(\Omega)$ is defined as the following equation

According to the research result from ^[2], it can be concluded that

Kirchhoff-type problems have a wide range of applications in mathematics and physics. In recent years, many scholars have devoted themselves to the study of Kirchhoff-type problems and have achieved many results ^[3,4,5,6]. Moreover, readers who are interested in this knowledge can refer to ^{[7,8,9,10,11,12,13,14,15]}. The research on the fractional $p$-Laplacian operator equation also achieved some results, such as ^{[16,17,18,19,20,21]}. Detailed knowledge about fractional-order related theories and differential inequalities can refer to ^{[22,23,24,25]} and the parabolic equation with logarithmic nonlinearity can refer to ^[26,27,28].

Until now, many scholars have studied the fractional Laplacian problem. In ^[1], the authors studied the fractional Kirchhoff-type problem.

where $[u]_{s}$ is the Gagliardo seminorm of $u$, let

where $s\in(0, 1)$, $q\in(2\theta, 2^*)$, $\theta\in[1, N/(N-2s)\big)$, $N > 2s$, $\Omega\subset \mathbb{R}^N$ 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:

where $h(x)$ is a sign-changing function on $\Omega$, $\lambda$ is a positive parameter, $M([u]^p_{s, p}) = [u]^{p(\theta-1)}_{s, p}$. 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 $\mathcal{L}_K$ 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 $1 < \alpha < N/(N-2s)$, 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).

2.   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 $W^{s, p}_0(\Omega)\hookrightarrow L^{\alpha p+\varrho}(\Omega), \varrho\in(0, p^*_s-{\alpha p}).$

The energy functional is

The Nehari functional is

And then, we define some sets as follows:

the mountain pass level

the Nehari manifold

Further, define the positive set and the negative sets as follows

From (2.2) and (2.3), we have

In order to facilitate the proof of the main results, next we give the necessary Lemmas and definitions.

Lemma $2.1$ Let $u\in W^{s, p}_0(\Omega)$ and $[u]_{s, p}\neq0$, $\rho\in (0, +\infty)$, then:

(ⅰ) $\lim\limits_{\rho\rightarrow0^+}E\big(\rho u(x)\big) = 0, \ \lim\limits_{\rho\rightarrow +\infty}E\big(\rho u(x)\big) = -\infty$;

(ⅱ) There is an unique value $\rho^*$, such that

(ⅲ) $E\big(\rho u(x)\big)$ is increasing on $(0, \rho^*)$, decreasing on $(\rho^*, +\infty)$ and takes the maximum at $\rho^*$;

(ⅳ) $I\big(\rho u(x)\big) > 0$ for all $0 < \rho < \rho^*$, $I\big(\rho u(x)\big) < 0$ for all $\rho > \rho^*$ and $I\big(\rho^* u(x)\big) = 0.$

Proof. (ⅰ) According to the known definition of $E\big(u(x)\big)$ on (2.2), i.e.

we have

we can conclude that (ⅰ) holds.

(ⅱ) Through simple calculations, we have

therefore, (ⅱ) holds.

(ⅲ) By the result in (ⅱ), we have

and

thus, (ⅲ) holds.

(ⅳ) According to the (2.11) and

thus

then, $I\big(\rho u(x)\big) > 0$ for all $0 < \rho < \rho^*$, $I\big(\rho u(x)\big) < 0$ for all $\rho > \rho^*$ and $I\big(\rho u(x)\big) = 0,$ (iv) holds.

Lemma $2.2$ ^[1,2] If $\varrho\in(0, p^*_s-{\alpha p})$, then:

Proof. Set $g(t) = \ln{t}-\frac{t^\varrho}{e\varrho}, \ t \in[1, +\infty)$. By a simple derivative calculation of the function, we get $g^{'}(t) = \frac{1}{t}-\frac{1}{e\varrho}t^{\varrho-1}$ and let $g^{'}(t) = 0$, then $t^* = e^\frac{1}{\varrho}$. Obviously $t^*$ is the maximum point of function $g(t)$, thus $g(t)\leq g(\ t^*) = 0$ for all $t \in[1, +\infty)$. This proves the above inequality.

Lemma $2.3$ If $\varrho\in(0, p^*_s-{\alpha p})$. Let $u\in W^{s, p}_0(\Omega)$ and $[u]_{s, p}\neq0$, we have

(ⅰ) If $0 < [u]_{s, p} < r(\varrho)$, then $I(u) > 0$;

(ⅱ) If $I(u)\leq0,$ then $[u]_{s, p}\geq r(\varrho).$

Proof. By Lemma 2.2 and the definition of Nehari function $I(u)$, we have

we can get

If $0 < [u]_{s, p} < r(\varrho)$, then

thus (ⅰ) holds.

Similarly, assume $I(u)\leq0,$ i.e.

thus (ⅱ) holds.

Lemma $2.4$ ^[2] If $s\in(0, 1)$, $1 < p < N/s$ holds, then the functionals $E(u)$ and $I(u)$ are well-defined and continuous on $W^{s, p}_{0}(\Omega)$. Moreover, $E(u)\in C^1(W^{s, p}_0(\Omega), \mathbb{R})$, and $I(u) = \langle E^{'}(u), u \rangle$ for all $u \in W^{s, p}_0(\Omega).$

Proof. Since

where $1\leq\alpha < p^*_s/p$, then we can claim that $E(u)$ and $I(u)$ are well-defined in $W^{s, p}_0(\Omega)$. Further, Similar to Lemma 2.3 in ^[2], one can prove the that $E(u)\in C^1(W^{s, p}_0(\Omega), \mathbb{R})$, and $I(u) = \langle E^{'}(u), u \rangle$ for all $u \in W^{s, p}_0(\Omega).$

Lemma $2.5$ ^[33] Let $s\in(0, 1)$ and $p\in[1, +\infty)$ such that $sp < N$. Let $q\in[1, p^*)$, $\Omega\subseteq \mathbb{R}^N$ be a bounded extension domain for $W^{s, p}(\Omega)$ and $\zeta$ be a bounded subset of $L^p(\Omega)$. Suppose that

then $\zeta$ is pre-compact in $L^q(\Omega).$

Lemma $2.6$ Let $s\in(0, 1)$, $1 < p < N/s$, $\alpha\in[1, p^*_s/p)$, $t\in[0, T)$, and assume that $u(x, t)$ is the solution of problem (1.1), then

Proof. By the definition of the weak solution, we have

So

Thus, we obtain

Definition $2.1$ A function $u(x, t)$ is called a weak solution of problem (1.1), where $(x, t)\in\Omega\times[0, T)$. If $u\in L^\infty\big(0, T; W^{s, p}_0(\Omega)\big)$ and $u_t\in L^2\big(0, T; L^2(\Omega)\big)$, for all $v\in W^{s, p}_0(\Omega)$, $t\in(0, T)$, the following equation holds

with $u(x, 0) = u_0(x)$ in $W^{s, p}_0(\Omega)$. Moreover, for $0\leq t < T$, the following inequality 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:

(ⅰ) If $u(t)$ exists for $0\leq t < +\infty$, then $T = +\infty$.

(ⅱ) If there exists a $t_0\in(0, +\infty)$ such that $u(t)$ exists for $0\leq t < t_0,$ but doesn't exist at $t = t_0$, then $T = t_0.$

Definition $2.3$ (Finite time blow-up) Let $u(x)$ be weak solution of problem (1.1), We call $u(t)$ a blow-up in finite time if the maximal existence time $T$ is finite and

3.   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.

Theorem $3.1$ Let $\alpha\in[1, p^*_s/p)$, $1 < p < N/s$. $u_0\in W^{s, p}_0(\Omega)$, If $E(u_0) < h$ and $I(u_0) > 0$, then problem (1.1) admits a global weak solution $u(t)\in L^\infty\big(0, T; W^{s, p}_0(\Omega)\big), u_t\in L^2\big(0, T; L^2(\Omega)\big)$ and $u(t)\in W$ for all $t\in[0, +\infty).$

Proof. By the definition of the weak solution, we take a set of smooth functions $\eta = \eta_k(x)\ (k = 1, 2, \cdot\cdot\cdot)$ such that $\{\eta_k(x)\}^\infty_{k = 1}$ is the basis function of $W^{s, p}_0(\Omega)$, and then construct a approximate solutions $u_m(t)$ of problem (1.1) as follows:

i.e. $u_m \ : \ [0, T)\longmapsto W^{s, p}_0(\Omega), \ u_m\in \text{span}\{\eta_1, \cdot\cdot\cdot\eta_m\}$, which satisfy

and

where $b_{km} = (u_m(0), \eta_k)$ are given constants, $k = 1, 2, \cdot\cdot\cdot, m.$ we use $(\cdot, \cdot)$ to represent the inner product in $L^2(\Omega)$.

Multiplying (3.1) by $d_{km}^{'}(t)$, then summing $k$ from $0$ to $m$, and finally integrating $t$ from $0$ to $t$, we get

According to (3.2), we obtain $d_{km}(0) = b_{km}$. Noticing $E(u_0) < h$, $I(u_0) > 0$, we have

and

then for a sufficiently large $m$, we can obtain

and

due to $W: = \{u\in W^{s, p}_0(\Omega)\ | \ I(u) > 0, \ E(u) < h\}\cup\{ 0 \},$ thus $u_m(x, 0)\in W$, this is true if $m$ is big enough.

Let $m\rightarrow+\infty, \ 0\leq t\leq T$, we prove $u_{m}(t)\in W$. Arguing by contraction, we suppose $u_{m}(t)\notin W$, for sufficiently large $m$, then exists a $t_0\in(0, T)$ satisfying $u_{m}(t_0)$ is on the boundary of $\partial W$, we have

thus can get $u_{m}(t_0)\in \mathcal{N}.$ And because $h = \inf\limits_{u\in N}E(u)$ which yields $E(u_{m}(t_0))\geq h,$ this is paradoxically. Then $u_m(t_0)\in W$ for $m\rightarrow+\infty$ and $0\leq t\leq T.$

By (3.3) and

we can get

For sufficiently large $m$, it yields

where $C_*$ is the best embedding constant for $W^{s, p}_0(\Omega)\hookrightarrow L^{\alpha p}(\Omega).$ Thus, we get $T = +\infty, \ u_m(t)\in W$ for sufficiently large $m$ and $0\leq t < +\infty$.

Next, we prove $u_m(t)\in L^\infty(0, T; W^{s, p}_0(\Omega))$ and $\ u_t\in L^2(0, T; L^2(\Omega))$. According to the (3.4) (3.5) and (3.6), we have $u_t\in L^2(0, T; L^2(\Omega))$, $u(t)\in L^\infty(0, T; W^{s, p}_0(\Omega))$. Based on the above proof, there exists a $u$ and subsequence $\{u_n\}$ and $\{u_m\}$ such that

By (3.7) and (3.11), we obtain

then $|u|^{\alpha p-2}u_n\ln|u_n|\rightarrow |u|^{\alpha p-2}u\ln|u|$ a.e. in $\Omega\times(0, +\infty).$

Since $u_n(t)\in W$ and

where $S_*$ is the imbedding constant for $W^{s, p}_0(\Omega)\hookrightarrow L^{(\alpha p+\varrho-1)\alpha p}(\Omega)$, which implies

In (3.1), we take a fixed $k$, let $m = n\rightarrow+\infty,$ then there is

and

where $v\in W^{s, p}_0(\Omega), \ t\in(0, +\infty)$, and by (3.2) gives $u(x, 0) = u_0(x)$ in $W^{s, p}_0(\Omega)$.

Finally, we prove the following inequality

In order to achieve the above result, we first consider

Since $\varrho\ln s \leq s^\varrho, \ s\in(0, +\infty), \ \varrho > 0,$ we have

By the weak lower semicontinuity and the above inequality, we obtain

which implies that the following inequality holds

Theorem $3.2$. Assume that $\alpha\in[1, p^*_s/p)$, $1 < p < N/s$ hold, and if $u_0\in W^{s, p}_0(\Omega)$, $E(u(0)) < h, \ I(u(0)) < 0,$ $u(t)$ is the weak solution of problem (1.1), then $u(t)$ blows up in a finite time $T$ ($T > 0$) and satisfies the following equation

In addition, if $E(u_0)\leq0$, then the solution $u(t)$ of problem (1.1) also satisfies

where $M(t) = \|u_0\|^{\frac{2-\alpha p}{2}}_2-\frac{4}{\alpha p-2} t |{\Omega}|^{\frac{2-\alpha p}{2}}(\alpha{ p})^{-1}.$

Proof. Let $u(t)$ be weak solution of problem (1.1), and $E(u_0) < h, \ I(u_0) < 0,$ we first define

then, we obtain

and

By (2.12), (3.14)and the following equation

we can get

Due to

square both sides of the above equation, we have

By (3.15), (3.16) and the Schwartz's inequality, we can get

i, e.

Now, we discuss the two cases $E(u_0)\leq0,$ and $0 < E(u_0) < h.$

(a) If $E(u_0)\leq0,$ then

Due to $I(u) < 0$ for all $t\in[0, T)$, so $Q^{''}(t) = -2I(u) > 0,$ $Q^{'}(t) > Q^{'}(0) = \|u_0\|^2_2 > 0$. Then for all $\delta\in[0, t-t_0),$ where $t_0\geq0, t\geq t_0$ and $Q^{'}(t) > 0,$ further we obtain

Moreover, we construct the following functional

By simple calculation and Lemma $5$, we have

and

Since $I(u) < 0$, thus

By $(2.10)$, $(2.12)$ and Holder's inequality, we obtain

where $\gamma(t) = f^{'}(t)+\int_\Omega|u_0|^2 dx$, and $\gamma^{'}(t)\geq 2 (\alpha p)^{-1}|\Omega|^{\frac{2-\alpha p}{2}}(\gamma(t))^{\frac{\alpha p}{2}}$, by the differential inequality, we obtain

where $M(t) = \|u_0\|^{\frac{2-\alpha p}{2}}_2-\frac{4}{\alpha p-2} t |{\Omega}|^{\frac{2-\alpha p}{2}}(\alpha{ p})^{-1}.$ By (3.18) (3.19), for sufficiently large $t$, we have

(b) If $0 < E(u_0) < h,$ by (3.18), ${\alpha p} > 2$ and Holder's inequality, then for a sufficiently large time $t$, we have

It implies for a sufficiently large time $t$, there exists a finite time $T > 0$ such that

Theorem $3.3$ If $\alpha\in[1, p^*_s/p)$, $1 < p < N/s$ hold, then there exists a $v\in\mathcal{N}$ that satisfies the equation $E(v) = \lim\limits_{u\in\mathcal{N}}E(u) = h$, and $v$ is the ground state solution of the following steady-state problem

Proof. By the definition of $h$ in (2.6), due to $u\in\mathcal{N},$ $I(u) = 0,$ we have

Let $\{u_n\}$ be a minimizing sequence, where $\{u_n\}\subset\mathcal{N}$ and $\lim\limits_{n\rightarrow+\infty}E(u_n) = h$, then

Since $\alpha\in[1, p^*_s/p)$, $1 < p < N/s$, it implies that $\{u_n\}$ is bound in $W^{s, p}_0(\Omega)$.

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

By Sobolev embedding theorem and logarithmic inequality, we have

where $\Omega^*\subset\Omega$ is a measurable subset, thus we can get that $|u_n|^{\alpha p}\ln|u_n|$ is uniformly bound. By integral convergence theorem and $|u_n|^{\alpha p}\ln|u_n|\rightarrow |v|^{\alpha p}\ln|v|$ a.e. in $\Omega$, we have

By Lemma $2.3(\text{ii})$ and $\{u_n\}\subset\mathcal{N},$ $I(u_n) = 0$, we can get

In other words

thus, we can conclude that $v\neq0.$

According to $(ii)$ of Lemma $2.1$, there exists a $\rho^* > 0$ satifying $\rho^*v\in\mathcal{N}, \ I(\rho^*v) = 0$, i.e.

Let

by $(iii)$ of Lemma $2.1$ and taking $I(v) < 0$, $[u(x)]^{\alpha p}_{s, p} < \int_\Omega|u|^{\alpha p}\ln|u(x)|dx$, we can be sure that $0 < \rho^* < 1.$

Then by the definition of $h$, the weak lower semicontinuity and (3.20), we obtain

and

By simply calculating, we can get

This contradicts our definition of $E(\rho^*v)\geq h$, so we can conclude $I(v) = 0, \ h = E(v) = \inf\limits_{u\in\mathcal{N}}E(u).$

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 $\xi\in\mathbb{R}$ such that

so we have

Since $I(v) = [v]^{\alpha p}_{\alpha p}-\int_\Omega|v|^{\alpha p}\ln|v|dx,$ then there is

Thus

Note that $I(v) = 0, \ [v]^{\alpha p}_{s, p} = \int_\Omega|v|^{\alpha p}\ln|v|dx$, then

thus, $\xi = 0, \ E^{'}(v) = 0.$ We get that $v$ is the ground state solution of the steady-state problem.

4.   Critical initial energy state $(E(u_0)=h)$

In this section, we will discuss the global existence and the finite time blow-up of weak solutions of problem (1.1) in the critical initial energy state $(E(u_0) = h)$.

Theorem $4.1$ Assume that $p\in(1, N/s), \ \alpha\in[1, p^*_s/p), \ u_0\in W^{s, p}_0(\Omega)$ and $E(u_0) = h, \ I(u_0)\geq0$ hold, then problem (1.1) admits a global weak solution $u(t)\in L^\infty(0, T; W^{s, p}_0(\Omega)), \ u_t\in L^2(0, T; L^2(\Omega))$ and $u(t)\in \overline{W}$ for all $t\in[0, +\infty).$

Proof. We take a function $\rho_m$ which satisfies $\rho_m > 0$ and $\lim\limits_{n\rightarrow+\infty}\rho_m = 1$. Let $u(x, 0) = u_{0m}(x) = \rho_mu_0(x), x\in\Omega$, for the following equations

Since $E(u_0) = h$, it implies $[u_0]_{s, p}\neq0.$ By Lemma $(2.1)$ and $I(u_0)\geq0,$ $0 < \rho_m < 1$, we obtain

and

By (iv) of lemma 2.1 $\big(\rho_m\frac{d}{d\rho_m}E(\rho_mu_0) = I(\rho_mu_0) \big)$ and

we have $I(\rho_mu_0) > 0.$ (i.e. $\rho_m\frac{d}{d\rho_m}E(\rho_mu_0) = I(\rho_mu_0) > 0$). Then $E(\rho_m u(x))$ is monotonically increasing on $(0, \rho^*)$, we can get

From Theorem 3.1, it follows that for each $m$, the above problem (4.1) admits a global weak solution $u_m(t)$, where $u_m(t)\in L^\infty(0, +\infty; W^{s, p}_0(\Omega)), \ u_{mt}\in L^{2}(0, +\infty; L^2(\Omega)),$ moreover $u_m(t)\in W$ for all $t\in[0, +\infty)$ and satisfying

in addition

By (4.2) and the following equation

we have

The remainder of the proof is similar to that in the proof of Theorem 3.1.

Theorem $4.2$ Let $u(t)$ be the weak solution of problem (1.1), assume that $p\in(1, N/s), \ \alpha\in[1, p^*_s/p), \ u_0\in W^{s, p}_0(\Omega)$ and $E(u_0) = h, \ I(u_0) < 0$ hold, then $u(t)$ blows up in a finite time $T_*.$

Proof. If $T_* = +\infty$, taking $Y(t) = \int^t_0\|u\|^2_2dr,$ we have

and

By (2.12), (4.3) and the following equation

we can get

Since

squaring both sides of the above equation, we have

By (4.4), (4.5) and the Schwartz's inequality, we can get

i, e.

Due to $I(u) < 0$ for all $t\in[0, T)$, so $Y{''}(t) = -2I(u) > 0$, $Y^{'}(t) > Y^{'}(0) = \|u_0\|^2_2 > 0$. For all $\delta\in[0, t-t_0),$ where $t_0\geq0, t\geq t_0$ and $Y^{'}(t) > 0,$ we obtain

Taking $|u| > 1$ and noting that ${\alpha p} > 2,$ we have

Moreover, by Holder's inequality, we have

By (4.6), (4.7), (4.8), for sufficiently large $t$, it implies that

and

So

which gives

It implies for a sufficiently large time $t$, there exists a finite time $T_*$ such that

and

This is contradictory to our assumption that $T_* = +\infty.$

Remark $1$. Theorems $3.1$ and $4.1$ show that $E(u_0)\leq h$ and $I(u_0) > 0$ holds, then problem (1.1) admits a global weak solution $u(t)\in L^\infty\big(0, T; W^{s, p}_0(\Omega)\big), \ u_t\in L^2\big(0, T; L^2(\Omega)\big)$ and $u(t)\in W$ for all $t\in[0, +\infty).$

Remark $2$. One can see from Theorems $3.2$ and $4.2$ that blow-up of the weak solutions can always occur if $E(u_0)\leq h, \ I(u_0) < 0$ holds.

5.   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.

Acknowledgments

This research was supported by the Project for Young Talents Growth of Guizhou Provincial Department of Education under(Grant No.Ky[2017]133), and by the project of Guizhou Minzu University under (Grant No.16yjrcxm002 and No.GZMU[2019]YB04).

Conflict of interest

The authors declare no conflict of interest.