Decay estimate and blow-up for fractional Kirchhoff wave equations involving a logarithmic source

1.   Introduction

Fractional differential equations can better describe practical problems than classical differential equations. This has attracted the interest and attention of many scholars to the fractional $p$-Laplacian $\left({ - \Delta } \right)_p^s$ ^[1,2,3,4,5]. The fractional 2-Laplacian operator of the form $\left({ - \Delta } \right)^s(p = 2)$ was first mentioned in physics when observing the Levy steady-state diffusion process, and later it was also used to depict abnormal plasma diffusion, fluid dynamics, and stochastic analysis ^[6,7,8]. Not only for mathematical purposes, but also for their importance in practical models, this paper will investigate the Kirchhoff-type wave models involving logarithmic nonlinearity and the fractional Laplacian operator as follows:

where, among them, $(-{\Delta})^s(s\in(0, 1))$ is the fractional Laplacian operator satisfying

The Kirchhoff term $M\left(\psi \right) = a + {\psi^{\theta - 1}}$ is a function that satisfies local Lipschitz continuity, $a > 0$, $1\leq \theta < \frac{{2_s^ * }}{2}$, where

is the critical exponent of the fractional Sobolev embedding inequality. If $\psi = \psi(t)$, we impose the following assumption on $M(\psi)$:

Moreover, $\Omega \subset {\mathbb{R}^N}({N \geq 1})$ is a bounded domain and the boundary $\partial \Omega$ is the smooth and nonlinear index $2\theta < k\leq {2_s^*}$.

Recently, the research on Kirchhoff-type equations ^[9,10] has received widespread attention. This kind of problem develops a major effect in the applications of nonlinear elasticity, electrorheological fluid, and image restoration ^[11,12]. It is meaningful to investigate the nonexistence, existence, blow-up, extinction, and decay estimation of its solutions. Kirchhoff ^[13] first introduced the following equation:

where $u$ denotes lateral displacement, $\delta$ denotes the resistance modulus, $\rho$ denotes mass density, $h$ represents the cross-section area, $P_0$ denotes initial axial tension, $L$ is the length, $E$ is Young's modulus, and $f$ represents external force. Since then, many researchers have become concerned with this kind of equation and have had excellent research results. In particular, many literatures have been devoted to discussing the Kirchhoff equation as follows:

where $f(u)$ is a nonlinear function that satisfies appropriate conditions and $M\geq 0$ is a local Lipschitz function. When $g(u_t) = \Delta u_t$, Wu and Tsai ^[14] made a profound study for Problem (1.3), where they found the upper bound of the blow-up time of solutions by the direct energy method. Yang and Han ^[15] also discussed Problem (1.3), through the Banach fixed point theorem, where they proved uniqueness as well as local existence of weak solutions. Then, through constructing a potential well, the lifespan of solutions with arbitrary initial energy was established. When $g(u_t)$ is the non-linear dissipation term ${\left| {{u_t}} \right|^{m - 1}}{u_t}$ or the linear dissipation term $u_t$, Ono studied Problem (1.3) involving $f\left(u \right) = {\left| u \right|^p}u$ in ^[16,17], when the initial energy is negative, and proved the finite time blow-up. In addition, when the initial energy was positive, he provided sufficient conditions for the finite time blow-up of the solutions. More research on the problems of Kirchhoff-type can be found in references ^{[18,19,20,21,22]}.

In 2017, Pan et al. ^[23] investigated the degenerate fractional Kirchhoff-type hyperbolic problems as follows:

Combining the potential wells theorem with the Galerkin method, they proved the global existence. Moreover, the vacuum isolating phenomenon and blow-up properties also were acquired. Additionally, the study of logarithmic source has a long history, appearing in different modules of physics ^[24].

Inspired by the above works, we investigate Problem (1.1) involving the fractional the Laplacian operator strong damping and logarithmic source, which is the first work that takes into account the blow-up property and decay estimate of weak solutions of Problem (1.1). We not only overcome the difficulty of logarithmic nonlinearity, but also deal with the fractional Laplacian operator. This work is extremely meaningful.

The structure of this article is as follows: In Section 2, we introduce important lemmas and basic definitions. In addition, potential wells and their properties are provided. Next, the local existence and uniqueness of the weak solutions are proved. Then we gain the global existence of weak solutions and establish an exponential decay estimate in Section 4. Finally, the finite time blow-up of the solutions is obtained.

2.   Notations and primary lemmas

We introduce some symbols, lemmas, and basic definitions in this section. For convenience, we define the $L^k(\Omega)$ norm through ${\left\| \cdot \right\|_k}$($1 \leq k \leq \infty$). First, some definitions of Sobolev space are reviewed, which can be found in ^[25].

Let the fractional exponent $s \in \left({0, 1} \right)$, ${H^s}\left({{\mathbb{R}^N}} \right)$ be the fractional Sobolev space satisfying

equipped with the norm

We denote space $\mathcal{O} = \mathcal{C}\left(\Omega \right) \times \mathcal{C}\left(\Omega \right) \subset {\mathbb{R}^N}$ where $\mathcal{C}\left(\Omega \right) = {\mathbb{R}^N}\backslash \Omega$, and then denote $Q = \left({{\mathbb{R}^N} \times {\mathbb{R}^N}} \right)\backslash \mathcal{O}.$ From nonlocal characteristics, we define the space

Let ${W_0} = \left\{ {u \in {W}:u\left({x, t} \right) = 0, x \in \mathcal{C}\left(\Omega \right) } \right\},$ which is a closed linear space, and ${W_0} \subset W$. Moreover, ${\left[ u \right]_s}$ is the Gagliardo seminorm satisfying

From the results of ^[26], it can be concluded that ${\left[ u \right]_s}$ is equivalent to the norm of $W_0$, and it is clear that the main space ${W_0}$ is a Hilbert space. Moreover, we denote the inner product in $L^2$ as ${\left({ \cdot, \cdot } \right)}$, the inner product in $W_0$ as ${\left({ \cdot, \cdot } \right)_{{W_0}}}$, and the dual product of $W_0$ in $Y_0$ as ${\langle { \cdot, \cdot } \rangle _{{W_0}}}$. $Y_0$ is the dual space of $W_0$.

For $u \in {W_0}$, we denote the main energy functional of this paper:

In addition, we define the potential energy functional

and Nehari functional

By direct computation, we have

and

We also define the depth of the potential well and the Nehari manifold, respectively, as

Further, we will introduce the sets

In this paper, to avoid confusion, we simply write $u(x, t)$ as $u(t)$ sometimes. Next, we give some definitions.

Definition 2.1. The function $u = u(x, t)$ is a weak solution of Problem (1.1) on $\Omega\times[0, T ]$, supposing that

and ${u_t} \in {L^2}\left({0, T;{W_0}} \right)$ satisfying $u(0) = u_0, u_t(0) = u_1$, and it holds that

for arbitrary $\phi \in W_0$, where the inner product

Definition 2.2. Let $u\left({x, t} \right)$ be a weak solution of Problem (1.1), and if the maximal existence time $T_{max}$ is finite and

we say that $u\left({x, t} \right)$ blows up in finite time.

Lemma 2.1. Let $u\in {W_0}\setminus\{{0}\}$, and we have

$(i) \; \mathop {\lim }\limits_{\lambda \to + \infty } J(\lambda u) = - \infty$, $\mathop {\lim }\limits_{\lambda \to {0^ + }} J(\lambda u) = 0$;

$(ii) \; J(\lambda u)$ is decreasing when $\lambda \in(\lambda_{\ast}, +\infty)$, and increasing when $\lambda \in(0, \lambda_{\ast})$;

$(iii) \; I(\lambda u) < 0$ when $\lambda \in (\lambda_{\ast}, +\infty)$, and $I(\lambda u) > 0$ when $\lambda \in (0, \lambda_{\ast})$.

Proof. By (2.6), we have

so the conclusion of $(i)$ is obviously valid. For the derivation of the above formula, we can obtain

Let

and since $k > 2\theta$ and $\theta\geq1$, we can obtain

Further, we have

where

which, together with $k > 2\theta\geq2$ and $\theta\geq1$, gives us $\mathop {\lim }\limits_{\lambda \to + \infty } h\left(\lambda \right) = - \infty$ and $\mathop {\lim }\limits_{\lambda \to 0} h\left(\lambda \right) = + \infty$. Taking the derivative of $h\left(\lambda \right)$, we obtain

So we infer that there is a unique $\lambda_0$ that satisfies $h\left(\lambda \right){|_{\lambda = {\lambda _0}}} = 0$, which means that

Combining (2.11) and (2.12), there is a unique $\lambda_1$ that satisfies $g\left(\lambda \right){|_{\lambda = {\lambda _1}}} = 0$. Then we can get that there is a $\lambda_* > \lambda_1$ satisfying $a\left\| u \right\|_{{W_0}}^2 + g\left(\lambda \right) = 0$, which means that $\frac{d}{{d\lambda }}J\left({\lambda u} \right){|_{\lambda = {\lambda _*}}} = 0$, $\frac{d}{{d\lambda }}J(\lambda u)$ is negative on $\left({{\lambda ^*}, + \infty } \right)$, and $\frac{d}{{d\lambda }}J(\lambda u)$ is positive on $\left({0, {\lambda ^*}} \right)$. Therefore, it can be seen that the conclusion of $(ii)$ is valid. By (2.6) and (2.7), we have

Thus, the conclusion of $(iii)$ holds. We have completed the proof of the properties of $J(\lambda u)$. □

Lemma 2.2. ^[27] Suppose that $\mu$ is a positive constant. We can get

and

where $e$ is a natural constant.

Lemma 2.3. ^[28] For $\forall r\in [1, 2_s^*]$ and $u \in W_0$, there is a constant $C_0(N, r, s) > 0$ that gives us

Lemma 2.4. ^[29] Assume that $W$ is a Banach space, and if $f \in L^k(0, T; W), \frac{\partial f}{\partial t}\in L^k(0, T; W)$, then when the value is transformed in a suitable set of measure zero in $[0, T]$, $f$ is a continuous injection from $[0, T]$ onto $W$.

Lemma 2.5. ^[30] Assume that $\left({X, d} \right)$ is a complete metric space, $F:X \to X$, and for any $x, y \in X$, we have

for some constant $0 < \delta < 1$. Then $F$ has a unique fixed point $\bar x \in X$ such that $F\left({\bar x} \right) = \bar x$.

Lemma 2.6. Assume that $u(x, t)$ is a weak solution of Problem (1.1), so the energy functional $E(t)$ is non-increasing about $t$.

Proof. We multiply the first equation of (1.1) by $u_t$ and integrate it on $\Omega\times[0, t)$, we can get

namely,

Deriving $E\left(t \right)$ about $t$, and we get

Therefore, the proof of the properties of $E(t)$ has been completed. □

3.   Local existence

Lemma 3.1. For any $2 \leq 2\theta < k\leq {2_s^*}$, $T > 0$, $u\in {\cal H} = C\left({\left[ {0, T} \right]; {W_0}} \right) \cap {C^1}\left({\left[ {0, T} \right]; {L^2}\left(\Omega \right)} \right)$, there is a unique

such that

Proof. $(i)$ Proof of the existence.

According to literature ^[31], there is an eigenfunction sequence ${\left\{ {{e_j}} \right\}_j} \subset C_0^\infty \left(\Omega \right)$ of fractional Laplacian operators, which is a completed orthogonal basis of $W_0$ and is an orthonormal basis in $L^2\left(\Omega \right)$. $\lambda_j > 0$ is defined as the corresponding eigenvalue satisfying $(-\Delta)^se_j = \lambda_je_j.$ Taking $W_m = Span\left\{ {{e _1, \cdots, e _m}} \right\}$ and constructing the approximate solutions

for every $\eta \in W_m$ and $t \geq 0$, satisfies the equations

For $j = 1, \cdots, m$, we make $\eta = e_j$ in the first equation in (3.2), and $\left\{ {h_{j}^m} \right\}_{j = 1}^m$ satisfies the Cauchy equations

where

which is a linear ordinary differential equation about $h_{j}^m$. On the basis of Peano's theorem, a local solution $h_j^m\in C^1[0, T ]$ has been obtained for the Cauchy problem mentioned above.

Now, we take $\eta = e_j$ and multiply two sides of the first equation (3.2) by $\dot h_{nj}^m\left(t \right)$, and then sum over $j$ from 1 to $m$ to get

We integrate the above equation on $[0, t]$ to get

Recalling $u\in {\cal H}$, we see that $\left\| u \right\|_{W_0}$ is bounded. Through the definition and assumption of function $M(m)$, we arrive at

Among them, $C_1$ is a positive number that only depends on $T$. Then we estimate the integral containing a logarithmic source on the right side of (3.3). Through Hölder's inequality, we can obtain

So next we deal with the term $\left\| {{v_{mt}}} \right\|_{{\frac{{2N}}{{N + 2s}}}}$ in (3.5). According to Lemma 2.3, we have

Then let ${\Omega _1} = \left\{ {x \in \Omega |\left| {{u_n}\left(x \right)} \right| < 1} \right\}$, ${\Omega _2} = \left\{ {x \in \Omega |\left| {{u_n}\left(x \right)} \right| \ge 1} \right\}$. Combining Lemmas 2.2 and 2.3, here we choose $0 < \mu\leq\frac{2N}{N-2s}-p$, and then we can obtain

so we can obtain

where ${C_2} = \left({\left[ {e\left({k - 1} \right)} \right]^{ { - {\frac{{2N}}{{N + 2s}}}}}}\left| \Omega \right|+{\left({e\mu } \right)^{{ - {\frac{{2N}}{{N + 2s}}}}}}C_{0}^{\frac{{2N\left({k - 1 + \mu } \right)}}{{N + 2s}}}C^{\frac{{2N\left({k - 1 + \mu } \right)}}{{N + 2s}}} \right)^{\frac{{N + 2s}}{{N}}}$.

Utilizing Young's inequality, then combining (3.6) and (3.8), (3.5) can be written as

Due to the convergence of $u_{0}^m$ and $u_{1}^m$, from (3.3), (3.4), and (3.9), we arrive at

where $\tilde C = \left\| {{u_1}} \right\|_2^2 + M\left({\left[ {{u_0}} \right]_s^2} \right)\left\| {u_0^m} \right\|_{{W_0}}^2 + {C_0^2C_2 }T > 0$ is independent of $m$. According to the definition of Kirchhoff function $M(m)$, we have

Combining (3.10) and (3.11), we have

Making use of the Gronwall inequality, we get

and integrating (3.13) on $[0, t]$, we arrive at

We substitute (3.14) into (3.10) to get

where ${C_T}$ is a normal number that depends on $T$. From (3.15), we get

By (3.16), (3.17), and the Aubin-Lions-Simon lemma ^[32], we can obtain

Therefore, $v_m(x, 0)$ makes sense, $v_m(x, 0) \rightarrow v(x, 0) \; in\; L^2(\Omega),$ and $v_m(x, 0) = u_{0}^m (x) \rightarrow u_0(x) \; in\; W_0.$ Thus, $v(x, 0) = u_0(x).$

Furthermore, dividing the two sides of the first equation in (3.2) by ${\left\| \eta \right\|_{{W_0}}}$, we have

By the $\rm{H\ddot{o}lder}$ inequality, (3.7), and (3.15), we get

For $\eta \in {W_0}\backslash \left\{ 0 \right\}$, upper bounds are simultaneously taken on both sides of Eq (3.20), and we have

namely

Combining $v_t \in L^\infty(0, T; L^2(\Omega))$ and $v_{tt}\in L^\infty(0, T; Y_0)$, through Lemma 2.4, we get

Thus $v_{mt} (x, 0)$ is meaningful and $v_{mt} (x, 0) \rightarrow v_t(x, 0) \; in\; Y_0.$ Owing to $v_{mt} (x, 0) = u_{1}^m (x) \rightarrow u_1(x)$ in $L^2(\Omega),$ we have that $v_t(x, 0) = u_1(x).$ We have completed the proof of the existence.

$(ii)$ Proof of the uniqueness.

Assuming Problem (1.1) has two solutions $v_1$ and $v_2$ with the same starting conditions, substituting them into Problem (3.1), and then, by subtracting the obtained two equations, we can get

Multiplying (3.23) by $v_{1t} - v_{2t}$ and integrating on $\Omega\times(0, T)$, we get

Obviously, this equality immediately yields $v_1 \equiv v_2$. This completes the proof. □

Based on the above lemma, we obtain the following theorem.

Theorem 3.1. Let ${u_0} \in W_0$, ${u_1} \in L^2(\Omega)$, and $2 \leq 2\theta < k\leq {2_s^*}$. Then there is a $T > 0$ that gives Problem (1.1) with a unique local solution $u(x, t)$ on $[0, T ]$ satisfying

Proof. For a given $T > 0$, we think over the important space ${\cal H} = C\left({\left[ {0, T} \right]; {W_0}} \right) \cap {C^1}\left({\left[ {0, T} \right]; {L^2}\left(\Omega \right)} \right)$ which has the following norm:

Let ${R^2} = M\left({\left[ {{u_0}} \right]_s^2} \right)\left\| {{u_0}} \right\|_{{W_0}}^2+\left\| {{u_1}} \right\|_2^2$, and then we denote

We first prove that ${{\cal{M}}_T}$ is a complete metric space. Let $\left\{ {{u_n}} \right\}$ be the Cauchy-Schwarz sequence in ${{\cal{M}}_T}$. Thus, for any $\varepsilon > 0$, there exists ${\mu _\varepsilon }$ such that if $n, m \ge {\mu _\varepsilon }$, then

and by the completeness of $L^2(\Omega)$ and $W_0$, there exist ${u} \in L^2(\Omega)$ such that $u_m \rightarrow u$ in $L^2(\Omega)$ and ${u} \in W_0$ such that $u_m \rightarrow u$ in $W_0$ when $m\rightarrow \infty$, namely

Therefore, ${{\cal{M}}_T}$ is a complete metric space.

Next, using the conclusion of Lemma 3.1, we denote $v = \Phi \left(u \right)$ for any $u \in {{\cal{M}}_T}$ as the unique solution to Problem (3.1). We will prove the mapping $\Phi$ is a contraction mapping satisfying $\Phi ({{\cal{M}}_T}) \subset {{\cal{M}}_T}$. We multiply the first equation of the Problem (3.1) with $v_t$ and integrate it on $\Omega \times \left({0, t} \right)$, and we obtain

Using a calculation method similar to the processes in (3.5) and (3.7), we find

Then by a similar computation to that of (3.4) and (3.14), we can derive that

Combining (3.25) and (3.26), (3.24) becomes

Further,

So we can choose a $T > 0$ to make it small enough so that $\left\| v \right\|_{\cal H}^2 \leq R^2.$

Next, we will prove that $\Phi$ is a contraction mapping. Let $v_1 = \Phi\left({{w_1}} \right)$, $v_2 = \Phi\left({{w_2}} \right)$ where ${w_1}, \; {w_2} \in {{\cal M}_T}$. Then if $v = v_1 - v_2$, $v$ satisfies

We will multiply the first equation of the above problem by $v_{t}$, and then integrate on $\Omega \times \left({0, t} \right)$, and we have

Next, we estimate the terms on the right side of (3.27) one by one. First, by performing calculations similar to (3.4), we obtain

Due to the function $M(m)$ being locally Lipschitz continuous, we arrive at

Next we use Young's inequality and Hölder's inequality to obtain

Similarly, taking advantage of the Lipschitz continuity of $f(x) = {\left| x \right|^{k - 2}}x\ln \left| x \right|$ in $\mathbb{R}^+ \rightarrow \mathbb{R}^+$, and from Lemma 2.3, we can see that

Substitute (3.28)–(3.30) into (3.27), and we can infer that

Applying the Gronwall inequality to the above inequality, we have

We can take $T > 0$ to make it small enough so that

and we conclude that there exists $\delta\in(0, 1)$ satisfying

Therefore, the above processes make certain the existence of weak solutions. □

4.   Global existence and decay estimate

First of all, we will prove the invariance of the set $W^+$.

Lemma 4.1. Let $u$ be the solution to Problem (1.1). In the case of $u_0 \in W^+$, $u_1 \in L^2(\Omega)$, and $0 < E(0) < d$, then for all $t\geq0$, we have $u(t)\in W^+$.

Proof. First, according to Lemma 2.6, we can get $E(t)\leq E(0) < d$. Next, we will prove through the method of contradiction that there is a minimum time $t_* \in (0, T_{max})$ that satisfies $u(t_*) \in \partial W^+$, i.e., $I\left({{u}\left({{t}} \right)} \right) > 0$ for $t \in [0, t_*)$ and $I\left({{u}\left({{t_*}} \right)} \right) = 0$. From (2.8), (2.9), and (2.13), we have

which means that $u(t_*)\ne 0$ and $u_t(t_*)\ne 0$. Thus according to (2.10), we get

which is not valid. Hence, we obtain $u(t)\in W^+$. □

Theorem 4.1. Let $u_0 \in W_0$, $u_1 \in L^2(\Omega)$, and $2 \leq 2\theta < k\leq {2_s^*}$. If $E(0) \leq d$, $I(u_0) > 0$, and then Problem (1.1) possesses a global weak solution $u \in L^{\infty}(0, +\infty; W_0)$ and ${u_t} \in {L^\infty }\left({0, + \infty; {L^2}\left(\Omega \right)} \right) \cap {L^2}\left({0, + \infty; {W_0}} \right)$. In addition, if

there exist positive constants $\kappa$ and $\gamma$ so that $E(t)$ satisfies the exponential decay estimate for $\forall\; t \geq0$ as follows:

Proof. We will prove the global existence and decay estimate.

Step 1. Global existence.

We will divide into two cases to prove the global existence.

Case 1: $I(u_0) > 0$ and $E(0) < d$

First of all, we need to declare that for the cases where $E(0) < d$ and $I(u_0) > 0$, we have:

$(i)$ If $I(u_0) > 0$ and $E(0) < 0$, this contradicts (2.8) and (2.9).

$(ii)$ If $I(u_0) > 0$ and $E(0) = 0$, by (2.8) and (2.9), it is evident that $u_0\equiv 0$ and $u_1\equiv 0$, which is an ordinary situation.

As a result, we only need to think about the cases where $0 < E({0}) < d$ and $I(u_0) > 0$.

From Lemma 4.1, we can see that $u(t) \in W^+$, which means that $I(u(t)) > 0$. Combining (2.5), (2.9), and (2.13), we get

namely

The right end of Eq (4.1) is a constant unrelated to $t$. This estimate enables us to take $T_{max} = +\infty$. As a consequence, we are able to get that Problem (1.1) has a unique global weak solution $u(t)$.

Case 2: $I(u_0) > 0$ and $E(0) = d$

Above all, we can select a sequence $\left\{ {{\theta _n}} \right\}_{n = 1}^\infty \subset (0, 1)$ satisfying $\mathop {\lim }\limits_{n \to \infty } {\theta _n} = 1$. Next we think about the problem as follows:

We claim that $I(u_{0n}) > 0$ and $0 < E(u_{0n}) < d$. In fact, from $I(u_0) > 0$ and ${\theta _n} \subset (0, 1)$, we get

On the other hand, combining the above inequality and Lemma 2.1, we have

which indicates that $J({\theta _n}u_0)$ is strictly increasing relative to ${\theta _n}$ and

Further, we have

Since $u_{0n} \rightarrow u_0$ and $u_{1n} \rightarrow u_1$ as $n \rightarrow +\infty$, we are able to get that there is a global weak solution $u(t)$ to the above problem through a method similar to Case 1.

Step 2. Decay estimate.

We establish an exponential energy decay estimate of Problem (1.1) when $I(u_0) > 0$ as well as $E\left(0 \right) \le \left\{ {d, \frac{{a\left({k - 2} \right)}}{{2p}}{{\left({\frac{{ae\mu }}{{C_0^{p + \mu }}}} \right)}^{\frac{2}{{k + \mu - 2}}}}} \right\}.$

It follows from (4.1) that

Next, we define an auxiliary function as follows:

where $\varepsilon > 0$ will be specified later. Through Lemma 4.1, we get $I(u(t)) > 0$. Then by (2.5), (2.8), and (2.9), we obtain

Next, utilizing Young's inequality, and combining (4.3) and Lemma 2.3, we arrive at

and

Taking $\xi$ small enough such that $\xi \le \frac{1}{{2C_0^2}}$, then we can obtain

and we fix $\xi$ and choose a sufficiently small normal number $\varepsilon$ such that $\varepsilon < 2\delta$, so (4.4) can be written as

Combining (4.4) and (4.6), it is easy to conclude that $L(t)$ is equivalent to $E(t)$, because there exist two constants $\eta_1 > 0$ and $\eta_2 > 0$ about $\varepsilon$ satisfying

Next, we derive $K(t)$ about $t$ and choose $0 < M < 2\theta$. From (2.5) and Lemma 2.3, we arrive at

Next we will estimate each term of (4.8). For the term $\left\| u \right\|_k^k$, by (4.2), we obtain

From (4.3) and Lemmas 2.2 and 2.3, for the logarithmic source term, we have

where $\mu$ satisfies $0 < \mu < 2_s^ * - k$. Substituting (4.9) and (4.10) into (4.8), we can derive that

where we require $E\left(0 \right) \le \left\{ {d, \frac{{a\left({k - 2} \right)}}{{2k}}{{\left({\frac{{ae\mu }}{{C_0^{k + \mu }}}} \right)}^{\frac{2}{{k + \mu - 2}}}}} \right\}$ such that

Now, we choose a small enough $M$ to make

We fix $M$ and then select a sufficiently small $\varepsilon$ to make

Through (4.11)–(4.13), we arrive at

Further, by (4.7), let $\gamma = \frac{\varepsilon M}{\eta_{2}}$ and (4.14) becomes

Eventually, by integrating (4.15) with $(0, t)$, we can infer that $K(t) \le K(0){e^{ - \gamma t}},$ and combining with (4.7), we can get

where $\kappa = \frac{L(0)}{{{\eta _1}}}$. □

5.   Blow-up

First, we will prove the invariance of the set $W^-$.

Lemma 5.1. Let $u(x, t)$ be a weak solution to Problem (1.1). Assuming $u_0 \in W^-$ and $E(0) < d$, for all $t\geq0$, we have $u(t)\in W^-$. In addition, we have the following inequality:

Proof. It follows from $u_0\in W^-$ that $I(u_0) < 0$. Next, we discuss it in two situations.

When $E(0)\leq0$, from (2.8) and (2.9), we arrive at

which means that $I(u(t)) < 0$, i.e., $u(t)\in W^-$.

When $0 < E(0) < d$, by (2.13), we have

which implies that $u(x, t)\neq0$. For $t\in[0, T_{max})$, we prove that $I(u(t)) < 0$. If not, there is a $t_1 \in (0, T_{max})$ to make $u(t_1)\in \partial W^-$, i.e., $I(u(t_1)) = 0$ and $I(u(t)) < 0$, $t\in[0, t_1)$. Looking back at (2.10), it is obvious that

which is opposite to (5.2). Therefore, for $t \in [0, T_{max}]$, we get $u(t)\in W^-$.

By Lemma 2.1, due to $I(u(t)) < 0$, there is a $\lambda_{\ast} < 1$ satisfying $I(\lambda_{\ast}u(t)) = 0$, and combining with (2.9) gives

Therefore, Lemma 4.1 is proven. □

Theorem 5.1. Let $2 \leq 2\theta < k\leq {2_s^*}$, $u_0 \in W^-$, and $u_1\in L^2(\Omega)$. Assuming $E(0) < d$, the solution of Problem (1.1) blows up in finite time.

Proof. First of all, by $E(0) < d$, $u_0\in W^-$, and Lemma 4.1, we get $u\in W^-$, which means that $I(u) < 0$.

We will prove that $u(t)$ blows up in finite time. If this is not established, we assume the global existence of $u$, i.e., $T_{max} = +\infty$. For any $T_0 > 0$, we define the positive function

We calculate the first-order and second-order derivatives of $Q(t)$, respectively, as

and

Through direct calculation, we arrive at

where the definition of $\psi(t)$ is as follows:

Using Hölder's inequality and the Cauchy-Schwarz inequality, for any $t \in (0, T_0)$, it is clear that $\psi(t) \ge 0$.

Further, (5.5) becomes

where $\varphi\left(t \right) = - k\left\| {{u_t}} \right\|_2^2 - 2I\left(u \right) - \left({k + 2} \right)\int_0^t {\left\| {{u_\tau }} \right\|_{{W_0}}^2d\tau }.$ Moreover, by (2.5), (2.13), and (5.1), we have

Since $E(0) < d$, we can get $\varphi \left(t \right) > 0$. The combination of (5.6) and (5.7) means that for all $t\in [0, T_0]$,

Let

and then we arrive at

As a result, it can be obtained that

namely

Therefore, the hypothesis does not hold. We have completed the proof. □

6.   Conclusions

We have investigated the initial boundary value problem that includes the fractional laplacian operator, strong damping term, and logarithmic source. To our knowledge, Kirchhoff proposed groundbreaking work on Kirchhoff-type problems, and we consider Problem (1.1), which has not been studied before. We proved the local existence by the contraction mapping principle and Faedo-Galerkin's method, namely, Theorem 3.1. Furthermore, global existence and properties of decay and blow-up when the initial value meets certain conditions are obtained, namely, Theorems 4.1 and 5.1. In future work, we will attempt to study the qualitative analysis of some interesting new models.

Author contributions

A. H. Sun: Methodology, writing–original draft, writing–review and editing; H. Xu: Methodology, writing–original draft. Both of authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank the editors and the anonymous referees for their careful reading and helpful suggestions which led to an improvement of this paper.

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.