Initial boundary value problem for fractional $p$-Laplacian Kirchhoff type equations with logarithmic nonlinearity

1.   Introduction

In this paper, we study the extinction and the blow-up for the following fractional $p$-Laplacian Kirchhoff type equations with logarithmic nonlinearity.

where

$\mathcal{Q} = \mathbb{R}^{2N}\backslash(\mathcal C\Omega\times \mathcal C\Omega)$, $\mathcal C\Omega = \mathbb{R}^N\backslash \Omega$, $\Omega\subset\mathbb{R}^N\ (N > 2s)$ is a bounded domain with Lipschitz boundary, $s\in(0, 1)$, $1 < p < 2$, $1 < q\leq 2$, $r > 1$, $\lambda, \ \beta > 0$, $(-\Delta)^s_p$ is the fractional $p$-Laplacian operator and satisfies

where $u(x)\in C^{\infty}$ and $u(x)$ has compact support in $\Omega$, $B_\gamma (x)\subset \mathbb{R}^N$ is the ball with center $x$ and radius $\gamma$. $u_0(x)\in L^\infty(\Omega)\cap W^{s, p}_0(\Omega)$ is a nonzero non-negative function, where $L^\infty(\Omega)$ and $W^{s, p}_0(\Omega)$ are Lebesgue space and fractional Sobolev space respectively, which will be given in section 2. $M(\cdot)$ is a Kirchhoff function with the following assumptions

($M_1$) $0 < s < 1, M(\tau): = a+b\theta\tau^{\theta-1}$ for $\tau\in\mathbb{R}^+_0: = [0, +\infty)$ ($a > 0, b\geq0$ are two constants), $\theta\geq1$;

($M_2$) $M: \mathbb{R}^+_0\rightarrow\mathbb{R}^+_0\backslash\{0\}$ is continuous and there exits $m_0\geq0$ such $M(\tau)\geq m_0$ for all $\tau\geq0.$

It is worth pointing out that the interest in studying problems like $(1.1)$ relies not only on mathematical purposes, but also on their significance in real models. For example, in the study of biological populations, we can use $u(x, t)$ to represent the density of the population at $x$ at time $t$, the term $(-\Delta)^s_pu$ represents the diffusion of density, $\mu|u|^{q-2}u$ represents the internal source and $\lambda|u|^{r-2}u\ln|u|$ denotes external influencing factors. For more practical applications of problems like $(1.1)$, please refer to the studies ^[1,2,3].

Compared with integer-order equations, it is very difficult to study the problem (1.1), which contains both non-local terms (including fractional $p$-Laplacian operators and Kirchhoff functions) and logarithmic nonlinearity. For the fractional order theory, we refer the readers to the studies ^[4,5,6]. In ^[7,8], the authors use Sobolev space and Nehari manifold to study the existence of solutions for fractional equations. In ^[9,10], the solutions for fractional equations are discussed by virtue of Nehari manifold and fibrillation diagram. By using different methods from above, the properties of the solutions for such partial differential equations are considered by the method of variational principle and topological theory in the the literature ^[11,12,13]. Moreover, the authors prefer to use potential well theory, Galerkin approximation and Nehari manifold method to prove the existence of solutions, decay estimation and blow-up, we refer the reader to the literature ^[14,15,16].

Existence, extinction and blow-up of solutions are three important topics which regard parabolic problems; in particular, the study of extinction properties has made great progress in recent years. In ^[17], Liu considered the following initial boundary value problem for the fractional $p$-Laplacian equation

where $1 < p < 2$, $q\leq1$ and $r, \lambda, \beta > 0$. By employing the differential inequality and comparison principle, they obtained the extinction and the non-extinction of weak solutions. In ^[18], Sarra Toualbia et al. considered the following initial boundary value problem of a nonlocal heat equations with logarithmic nonlinearity

where $p\in(2, +\infty)$. By using the logarithmic Sobolev inequality and potential well method, they obtained decay, blow-up and non-extinction of solutions. In ^[19], Xiang and Yang studied the first initial boundary value problem of the following fractional $p$-Kirchhoff type

where $M: [0, \infty)\rightarrow(0, \infty)$ is a continuous function, $0 < s < 1 < p < 2$, $1 < q\leq 2$, $r > 1$, $\lambda, \mu > 0$. Under suitable assumptions, they proved the extinction and non-extinction of solutions and perfected the Gagliardo-Nirenberg inequality. For more information on the extinction properties of the solution, please refer to the studies ^{[20,21,22,23]}.

Inspired by the above works, we overcome the research difficulties of logarithmic nonlinearity, $p$-Laplace operator and Kirchhoff coefficients in problem $(1.1)$ based on the potential well theory, Nehari manifold and differential inequality methods, we give the extinction and the blow-up properties of solutions. In addition, we give the global boundedness of the solution by appropriate assumptions. To the best of our knowledge, it is the first result in the literature to investigate the extinction and blow-up of solutions for fractional $p$-Laplacian Kirchhoff type with logarithmic nonlinearity.

In order to introduce our main results, we first give some related definitions and sets.

Definition $1.1$(Weak solution). A function $u(x, t)$ is said to be a weak solution of problem (1.1), if $(x, t)\in\Omega\times[0, T)$, $u\in L^p\big(0, T;W^{s, p}_0(\Omega))\cap C(0, T; L^2(\Omega)\big)$, $u_t\in L^2\big(0, T;L^2(\Omega)\big)$, $u(x, 0) = u_0(x)\in W^{s, p}_0(\Omega)$, for all $v\in W^{s, p}_0(\Omega)$, $t\in(0, T)$, the following equation holds

where

Definition $1.2$(Extinction of solutions). Let $u(t)$ be a weak solution of problem (1.1). We call $u(t)$ an extinction if there exists $T > 0$ such that $u(x, t) > 0$ for all $t\in(t, T)$ and $u(x, t)\equiv0$ for all $t\in[T, +\infty)$.

Define the following two functionals on $W^{s, p}_0(\Omega)$

Let

Remark $1$ Since $u\in Z$, $1 < p < 2$, $M(\cdot)$ is a continuous function and

where $0 < \sigma < p^*_s -r$, then we can claim that $E(u)$ and $I(u)$ are well-defined in $W^{s, p}_0(\Omega)$. Further, by arguing essentially as in ^[24], one can prove the that

is continuous from $W^{s, p}_0(\Omega)$ to $\mathbb{R}$. It follows that $E(u)$ and $I(u)$ are continuous.

Define some sets as follows

the mountain pass level

the Nehari manifold

Moreover, we define

Let $\lambda_1$ be the first eigenvalue of the problem

and $\phi(x) > 0$ a.e. in $\Omega$ be the eigenfunction corresponding to the eigenvalue $\lambda_1 > 0$, $\phi(x)\in L^\infty(\Omega)$ and $\|\phi\|_{L^\infty(\Omega)}\leq1.$

First, we give some results satisfying $I(u_0) > 0$ and $q = 2.$

Theorem $1.1$ Assume that $I(u_0) > 0$, $r = p$ and $q = 2.$ Let $m_0$ be as in assumption $(M_2)$, and let

where $L(p, \Omega)$ and $R$ are given in Lemma 2.1 and Lemma 2.5. Then, there exist positive constants $C_1$, $C_2$, $T_1$ and $T_2$ such that

(ⅰ) If $\lambda < \lambda_1P_1$, then the weak solution of $(1.1)$ vanishes in the sense of $\|\cdot\|_2$ as $t\rightarrow+\infty.$

(ⅱ) If $2N/(N+2s) < p < 2$ and $\lambda < \lambda_1P_1$ or $1 < p\leq 2N/(N+2s)$ and $\lambda < P_2$, then the nonnegative solutions of $(1.1)$ vanish in finite time, and

for $2N/(N+2s) < p < 2$, and

for $1 < p < 2N/(N+2s)$.

Theorem $1.2$ Assume that $I(u_0) > 0$, $0 < \sigma\leq p^*_s-r$, $r > p$ and $q = 2$. Let $m_0$ be as in assumption $(M_2)$, and let

where

$C_{p^*_s}$ is the best constant of embedding from $W^{s, p}_0(\Omega)$ to $L^{p^*_s}(\Omega)$, $L(p, \Omega)$ and $R$ are given in Lemma 2.1 and Lemma 2.5, and $\varepsilon, \varepsilon_1 > 0$ are two constants. Then, there exist positive constants $C_4$, $C_5$, $C_6$, $C_7$, $T_3$ and $T_4$ such that the non-negative weak solution of problem $(1.1)$ vanishes in finite time and

for $2N/(N+2s)\leq p < 2$, $\lambda < P_3$, and

for $1 < p < 2N/(N+2s)$, $\lambda < P_4$.

Secondly, we give some results satisfying $I(u_0) > 0$ and $q < 2.$

Theorem $1.3$ Assume that $I(u_0) > 0$, $p = r$, $l_1 > l\geq1$ and $1 < q < 2$. Let $m_0$ be as in assumption $(M_2)$, and let

where $L(p, \Omega)$ and $R$ are given in Lemma 2.1 and Lemma 2.5. If $2N/(N+2s) < p < 2$ with $\lambda < \lambda_1P_1$ or $1 < p\leq2N/(N+2s)$ with $\lambda < P_5$, then the non-negative weak solution of problem $(1.1)$ vanishes in finite time for any non-negative initial data.

Theorem $1.4$ Assume that $I(u_0) > 0$, $l_1 > l\geq1$, $r\leq2$ and $1 < q < 2$. Let $m_0$ be as in assumption $(M_2)$, and let

and $\varepsilon_2$, $\varepsilon_3 > 0$ are two constants. If $2N/(N+2s) < p < 2$ with $\lambda < P_6$ or $1 < p\leq2N/(N+2s)$ with $\lambda < P_7$, then the non-negative weak solution of problem $(1.1)$ vanishes in finite time for any non-negative initial data.

Finally, we discuss the global boundedness and blow up of weak solutions.

Theorem $1.5$ Let $u(x, t)$ be the weak solution of problem $(1.1)$.

(ⅰ) If $E(u_0) < 0$, $r = p > q$ and $\theta = 1$, then the weak solution $u(x, t)$ blows up at $+\infty$;

(ⅱ) If $0 < E(u_0)\leq h$ and $I(u_0)\geq0$, then the weak solution $u(x, t)$ is globally bounded.

The rest of the paper is organized as follows. In Section 2, we give some related spaces and lemmas. In Section 3, we give the proof process for the main results of problem $(1.1)$.

2.   Preliminaries

In order to facilitate the proof of the main results, we start this section by introducing some symbols and Lemmas that will be used throughout the paper.

In this section, we assume that $0 < s < 1 < p < 2$ and $\Omega\in\mathbb{R}^N\ (N > 2s)$ is a bounded domain with Lipschitz boundary. We denote by $\|u\|_i \ (i\geq1)$ the norm of Lebesgue space $L^{i}(\Omega)$. Let $W^{s, p}(\Omega)$ be the linear space of Lebesgue measurable functions $u$ from $\mathbb{R}^{N}$ to $\mathbb{R}$ such that the restriction to $\Omega$ of any function $u$ in $W^{s, p}(\Omega)$ belongs to $L^p(\Omega)$ and

where $\mathcal{Q} = \mathbb{R}^{2N}\backslash(\mathcal C\Omega\times \mathcal C\Omega)$, $\mathcal C\Omega = \mathbb{R}^N\backslash \Omega$. The space $W^{s, p}(\Omega)$ is equipped with the norm

We further give a closed linear subspace

As shown in ^[19], it can be concluded that

is an equivalent norm of $W^{s, p}_0(\Omega)$.

Next we give the necessary Lemmas.

Lemma $2.1$ (^[25]) Let $u\in W^{s, p}_0(\Omega)\backslash \{0\}$. Then

where $L(p, \Omega): = \frac{|\Omega|}{ep}+\frac{1}{e(p^*_s-p)}C^{p^*_s}_{p^*_s}$, $C_{p^*_s}$ is the best constant of embedding from $W^{s, p}_0(\Omega)$ to $L^{p^*_s}(\Omega).$

Lemma $2.2$ (^[26]) Let $y(t)$ be a non-negative absolutely continuous function on $[T_0, +\infty)$ satisfying

where $\alpha, \beta > 0$ are constants and $k\in(0, 1)$. Then

where $T_* = \frac{1}{(1-k)\beta}\ln\big(1+\frac{\beta}{\alpha}y^{1-k}(T_0)\big).$

Lemma $2.3$ (^[27]) Suppose that $\beta^*\geq0$, $N > sp\geq1$, and $1\leq r\leq q \leq(\beta^*+1)p^*_s$, then for $u$ such that $|u|^{\beta^*}u\in W^{s, p}_0(\Omega)$, we have

with $\vartheta = \frac{[(\beta^*+1)p^*_s-q]r}{[(\beta^*+1)p^*_s-r]q}$, where $C_{p^*_s}$ is the embedding constant for $W^{s, p}_0(\Omega)\hookrightarrow L^{p^*_s}(\Omega).$

Lemma $2.4$ (^[19]) Let $1 < p < \infty$ and $g: \mathbb{R}\rightarrow\mathbb{R}$ be an increasing function. Define

then

Lemma $2.5$ Assume that ($M_1$) holds. Let $u\in W^{s, p}_0(\Omega)\backslash\{0\}$, $0 < \sigma\leq p^*_s-r$. We have

(ⅰ) if $0 < \|u\|\leq R$, then $I(u) > 0$;

(ⅱ) if $I(u)\leq0$, then $\|u\| > R$,

where

$\ C_{r+\sigma}$ is the embedding constant for $W^{s, p}_0(\Omega)\hookrightarrow L^{r+\sigma}(\Omega).$

Proof. Since $u\in W^{s, p}_0(\Omega)\backslash\{0\}$, and

Then by the definition of $I(u)$, we obtain

where $C_{r+\sigma}$ is the embedding constant for $W^{s, p}_0(\Omega)\hookrightarrow L^{r+\sigma}(\Omega).$

We can get

If $0 < \|u\|\leq R$, then it follows from the definition of $R$ that

thus (ⅰ) holds.

If $I(u)\leq0$, by $(1.12)$, we have

thus (ⅱ) holds.

Lemmas 2.6 is similar to [28,Lemmas 9], so we ignore its proof.

Lemma $2.6$ (^[28]) Assume that $E(u_0)\leq h$, then the sets $\mathcal{N_+}$ and $\mathcal{N_-}$ are both invariant for $u(t)$, i.e, if $u_0\in \mathcal{N_-}$ (resp. $u_0\in\mathcal{N_+}$), then $u(t)\in \mathcal{N_-}$ (resp. $u(t)\in\mathcal{N_+}$) for all $t\in[0, T).$

Lemma $2.7$ (^[29]) Let $\alpha$ be positive. Then

3.   Proof of main results

In this section, we prove that the main results of problem $(1.1)$.

Proof of Theorem 1.1

(1) Taking $v = u$ in Definition 1.1, we obtain

By Lemma 2.1, Lemma 2.5 and Lemma 2.6, we obtain

Since $\lambda < \lambda_1P_1$, we get

thus

which implies that $\|u(\cdot, t)\|_2\rightarrow0$ as $t\rightarrow+\infty.$

(2) We consider first the case $2N/(N+2s) < p < 2$ with $\lambda < \lambda_1P_1.$ By (3.1) and Lemma 2.1, Lemma 2.5 and Lemma 2.6, we have

Using Hölder's inequality and the fractional Sobolev embedding theorem, we have

where $C_{p^*_s} > 0$ is the embedding constant. By $(3.2)$, $(3.3)$ and $\lambda < \lambda_1P_1$, we obtain

where

Setting $y(t) = \|u(\cdot, t)\|^2_2$, $y(0) = \|u_0(\cdot)\|^2_2$, by Lemma 2.2, we obtain

where

Next, we consider the case $1 < p\leq2N/(N+2s)$ and $\lambda < \lambda_1P_2.$ Taking $\nu = u^l$ in Definition 1.1, where $l = \frac{2N-(s+N)p}{sp}\geq1$, by Lemma 2.1, Lemma 2.4, Lemma 2.5 and Lemma 2.6, we obtain

where $G = \Big(\frac{m_0lp^p}{(p+l-1)^p}-\frac{\lambda pL(p, \Omega)}{p+l-1}-\frac{\lambda p\ln(R)}{\lambda_1(p+l-1)}\Big)$. By the very choice of $l$ and the fractional Sobolev embedding theorem, we have

Hence,

where

since $\lambda < P_2$, then $C_2 > 0$. Setting $y(t) = \|u(\cdot, t)\|_{l+1}$, $y(0) = \|u_0(\cdot)\|_{l+1}$, by Lemma 2.2, we obtain

where

The proof is completed.

Proof of Theorem 1.2

We consider first the case $p < r < 2$ and $2N/(N+2s) < p < 2$. Taking $\nu = u$ in Definition 1.1, we have

By Lemma 2.1, Lemma 2.5 and Lemma 2.6, we obtain

Using Hölder's inequality and the interpolation inequality, the fractional Sobolev embedding theorem and Young inequality, we can easily obtain (see ^[19])

where $s_1 > r$, $\vartheta_1\in(0, 1)$, $\varepsilon > 0$ and

By $(3.13)$ and $(3.14)$, we have

where

Since $2N/(N+2s) < p < 2$ and $\lambda < P_3$, by the fractional embedding theorem and $(3.3)$, we obtain

and $C_3 > 0, \; C_4 > 0$, where

Similar to the Theorem 1.1, one can prove the that

where

When $1 < p\leq2N/(N+2s)$, $p < r\leq2$ and $\lambda < P_4$. Taking $\nu = u^l\ (l = \frac{2N-sp-Np}{sp}\geq1)$ in Definition 1.1, by Lemma 2.4 and Lemma 2.7, we obtain

further, we have

where $r_1 = \frac{p(l+1)}{l+p-1}$, $r_2 = \frac{p(l+r+\sigma-1)}{l+p-1}$. Note that, since $r_1 < p^*_s$, by the Hölder's inequality and the fractional Sobolev embedding theorem, we have

Using the same discussion as above, one can conclude that

where $s_2 > r_2$, $\vartheta_2\in(0, 1)$, $\varepsilon_1 > 0$ and

Combining $(3.22)$-$(3.24)$ and $\lambda < P_4$, we obtain

and $C_6 > 0$, $C_7 > 0$, where

Using Lemma 2.2 and a direct calculation, we have

where

The proof is completed.

Proof of Theorem 1.3

We consider first the case $2N/(N+2s) < p < 2$ with $\lambda < \lambda_1P_1$. Taking $v = u$ in Definition 1.1, we obtain

Note that $\|u\|^p\geq\lambda_1\|u\|^p_p$ and by Lemma 2.1, Lemma 2.5 and Lemma 2.6, we have

By $(3.3)$ and $\beta > 0$ and $\lambda < \lambda_1P_1$, we obtain

where

By a direct calculation, we obtain

where

Next, we consider the case $1 < p\leq2N(N+2s)$ with $\lambda < P_2$. Multiplying $(1.1)$ by $u^{l_1}\ (l_1 > l\geq1)$ and integrating, by Lemma 2.1, Lemma 2.4, Lemma 2.5 and Lemma 2.6, we obtain

where $G_1 = \Big(\frac{m_0l_1p^p}{(p+l_1-1)^p}-\frac{\lambda pL(p, \Omega)}{p+l_1-1}-\frac{\lambda p\ln(R)}{\lambda_1(p+l_1-1)}\Big)$. By Lemma 2.3, we obtain

where

By the choice of $l_1$, we have $0 < \vartheta_3 < 1$. Hence, using the Young inequality, for every $r_3 > 0$ and $\varepsilon_2 > 0$, we obtain

We now choose

and we notice that $\frac{\vartheta_3r_3(p+l_1-1)}{(p+l_1-1)-(1-\vartheta_3)r_3} = q+l_1-1$. That means

We choose

By $(3.34)$ and $(3.37)$ and $\lambda < P_5$, we obtain

where

and $C_9 > 0,$ which implies that

where

The proof is completed.

Proof of Theorem 1.4

We consider first the case $r < 2$. When $2N/(N+2) < p < 2$, multiplying $(1.1)$ by $u$, we obtain

Similar to the Theorem 1.2, we obtain

Taking $\beta^* = 0$ in Lemma 2.3, we have

where $\vartheta_4 = \frac{(p^*_s-2)p}{(p^*_s-q)2}$. Then, taking into account that $\vartheta_4\in(0, 1)$, we can apply Young's inequality: for every $r_4 > 0$ and $\varepsilon_3 > 0$, we have

Taking

then $\frac{\vartheta_4r_4p}{p-k_2(1-\vartheta_4)} = q$. By $(3.39)$, $(3.41)$, Lemma 2.5 and $\lambda < P_6$, and let $\varepsilon_3 = \Big(\frac{m_0-\lambda L(r, \Omega)R^{r-p}}{\beta}\Big)^{\frac{p-(1-\vartheta_4)r_4}{p}}$, we obtain

where

which implies that our result holds.

When $1 < p\leq2N/(N+2s)$, we multiply $(1.1)$ by $u^{l_1}\ (l_1 > l\geq1)$, by Lemma 2.4 and Lemma 2.7, we obtain

where $r_5 = l_1+r+\sigma-1$. Using the Holder inequality and $(3.37)$, and we choose

then, we have

for $0 < \sigma < 2-r$. After performing some simple calculations, we finally obtain

where

Note that $\lambda < P_7$, then $C_{11} > 0,$ which implies that

where

The proof is completed.

Proof of Theorem 1.5

(ⅰ) By the definition of $E(u)$ and $I(u)$, we obtain

Choosing $\nu = u_t$ in Definition 1.1, we have

Note that

By $(3.47)$ and $(3.48)$, we obtain

which implies that

Setting $\Gamma(t) = \frac{1}{2}\int_\Omega|u(x, t)|^2dx$, then we have

By $(3.46)$ and $(3.49)$ and $(3.50)$, we obtain

Since $p > q$, then $\frac{1}{q}\mu(p-q)\|u\|^q_q > 0$, we obtain

By a simple calculation, we get

which implies that our result holds.

(ⅱ) Here, we only prove the case of $E(u_0) < h$, and the proof of $E(u_0) = h$ is similar. Choosing $\nu = u$ in Definition 1.1, we have

namely

Next, taking $\nu = u_t$ in Definition 1.1 and integrating with respect to time from $0$ to $t$, we have

We claim that $u(x, t)\in W$ for any $t > 0$. If it is false, there exists a $t_0\in\mathbb{R}^+_0\backslash\{0\}$ such that $u(t_0)\in\partial W$, which implies

From $(3.53)$, $E(u(t_0)) = h$ is not true. So $u(t_0)\in\mathcal{N}$, then by the definition of $h$ in $(1.9)$, we have $E(u(t_0))\geq h$, which also contradicts with $(3.53)$. Hence, $u(t_0)\in W$. By $(3.52)$ and $u(t)\in W$ for all $t > 0$, we obtain

Remark $2$ Compared with problem (1.4), we not only discuss the extinction of weak solutions of problem (1.1) with logarithmic nonlinearity, but also prove that the weak solutions are globally bounded and blow up at infinity.

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 potential conflict of interests.