Genetic etiology of cleft lip and cleft palate

1.   Introduction

In this paper, we deal with the following plate equation with nonlinear damping and a logarithmic source term

where $\Omega$ is a bounded domain in $\mathbb{R}^n$ ($n\geq1$) with sufficiently smooth boundary $\partial\Omega$, $\nu$ is the unit outer normal to $\partial\Omega$ and $k$ is a positive real number, $u_0(x),$ and $u_1(x)$ are given initial data. The parameter $m\geq2$ and $p$ satisfies

The logarithmic nonlinearity is of much interest in many branches of physics such as nuclear physics, optics and geophysics (see [5,6,15] and references therein). It has also been applied in quantum field theory, where this kind of nonlinearity appears naturally in cosmological inflation and in super symmetric field theories [4,13].

Let us review somework with logarithmic term which is closely related to the problem (1). Birula and Mycielski[6,7] studied the following problem

which is a relativistic version of logarithmic quantum mechanics and can also be obtained by taking the limit $p\rightarrow1$ for the $p$-adic string equation [16,36]. Cazenave and Haraux [8] established the existence and uniqueness of a solution to the Cauchy problem for the following equation

in $\mathbb{R}^{3}$. Using some compactness method, Gorka [15] established the global existence of weak solutions for all $(u_0, u_1)\in H_0^1\times L^2$ to the initial boundary value problem of equation (4) in the one-dimensional case. In [5], Bartkowski and Gorka obtained the existence of classical solutions and investigated weak solutions for the corresponding Cauchy problem of equation (4) in the one-dimensional case. Recently, using potential well combined with logarithmic Sobolev inequality, Lian et al. [25] derived the global existence and infinite time blow up of the solution to the initial boundary value problem of (4) in finite dimensional case under suitable assumptions on initial data. Similar results were obtained by Lian et al. [26] for nonlinear wave equation with weak and strong damping terms and logarithmic source term. Hiramatsu et al. [19] also introduced the following equation

to study the dynamics of Q-ball in theoretical physics. A numerical research was given in that work, while, there was no theoretical analysis for this problem. For the initial boundary value problem of(5), Han [17] obtained the global existence of weak solution in $\mathbb{R}^{3}$, and Zhang et al. [40] obtained the decay estimate of energy for the problem (5) in finite dimensional case. Later, the authors in [20] considered the initial boundary problem of (5) in $\Omega \subset \mathbb{R}^{3}$, they proved that the solution will grow exponentially as time goes to infinity if the solution lies in unstable set or the initial energy is negative; the decay rate of the energy was also obtained if the solution lies in a smaller set compared with the stable set. Peyravi[35] extended the results obtained in [20] to the following logarithmic wave equation

Recently, Al-Gharabli and Messaoudi [1] considered the following plate equation with logarithmic source term

where $\Omega$ is a bounded domain in $\mathbb{R}^2$, and obtained the global existence and decay rate of the solution using the multiplier method. As the special case, i.e. $h(u_t) = u_t$ in (6), the same authors [2] established the global existence and the decay estimate by constructing a Lyapunov function. Moreover, Al-Gharabli et al. [3] considered the following initial boundary value problem of viscoelastic plate equation with with logarithmic source term

they established the existence of solutions and proved an explicit and general decay rate result. However, there is no information on the finite or infinite blow up results in these researches [1,2,3].

At the same time, there are many results concerning the existence and nonexistence on evolution equation with polynomial source term. For example, for plate equation with polynomial source term $|u|^{p-2}u$, Messaoudi [31] considered the following problem

established an existence result and showed that the solution continues to exist globally if $m\geq p$ and blows up in finite time if $m < p$ and the initial energy is negative. This result was later improved by Chen and Zhou [12], see also Wu and Tsai [37]. Here, we also mention that there are a lot of results on the global well-posedness of solutions to the initial boundary value problem of nonlinear wave equations can be found [30,39] and papers cited therein by using of potential well method.

To the best of our knowledge, there are few results on the evolution equation with the nonlinear logarithmic source term $|u|^{p-2}u\log |u|^k$ ($p>2$). Kafini and Messaoudi [22] studied the following delayed wave equation with nonlinear logarithmic source

obtained the local existence by using the semigroup theory and proved a finite time blow-up result when the initial energy is negative. Of course, these results also hold for the equation (8) without delay term (i.e. $\mu_2 = 0$). However, there are no results on general decay and blow-up for positive initial energy. Chen et al. [9,10] studied parabolic type equations with logarithmic nonlinearity $u\log |u|^k$, and obtained the global existence of solution and the solutions cannot blow up in finite time. Recently, Chen and Xu [11] study the initial-boundary value problem for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity and obtain the similar results. Nhan and Truong studied parabolic $p$-Laplacian equation [23] and pseudo-parabolic $p$-Laplacian equation [24] with logarithmic nonlinearity $|u|^{p-2}u\log |u|^k$ where they need the $p$-Laplacian term to control the logarithmic nonlinearity. We also refer to [18], where pseudo-parabolic $p$-Laplacian equation with logarithmic nonlinearity $|u|^{q-2}u\log |u|^k$ was considered.

Motivated by the above mentioned papers, our purpose in this research is to investigative the existence, energy decay and finite time blow-up of the solution to the initial boundary value problem (1). We note here that (i) the term $u$ plays an important role in the studying the problem (6)(see [1,2]) and (7) (see [3]) when the Logarithmic Sobolev inequality is used, while we do not care the term $u$ in this paper; (ii) The constant $k$ in (6) and (7) should satisfy $0<k<k_0$, where $k_0$ is defined by $\frac{2 \pi}{k_{0} c_{p}} = e^{-3-\frac{2}{k_{0}}}$ (see details in [1,2,3]), while we only need $k>0$.

The rest of this article is organized as follows: Section 2 is concerned with some notation and some properties of the potential well. In Sect. 3, we present the existence and uniqueness of local solutions to (1) by using the contraction mapping principle. In Sect. 4, we prove the global existence and energy decay results. The proof of global existence result is based on the potential well theory and the continuous principle; while for energy decay result, the proof is based on the Nakao's inequality and some techniques on logarithmic nonlinearity. In Sect. 5, we prove the the finite time blow-up when the initial energy is negative. In Sect. 6, we establish the finite time blow-up result for problem (1) with $m = 2$ under the arbitrarily high initial energy level ($E(0)>0$).

2.   Preliminaries

We give some material needed in the proof of our results. We use the standard Lebesgue space $L^p(\Omega)$ and Sobolev space $H^2_0(\Omega)$ with their usual scalar products and norms. In particular, we denote $\| .\| = \| .\|_{2}$. By Poincar$\acute{e}$ 's inequality [27], we have that $\|\Delta \cdot\|$ is equivalent to $\|\cdot\|_{H_0^2}$ and we will use $\|\Delta \cdot\|$ as the norm of $\|\cdot\|_{H_0^2}$, the corresponding duality between $H_0^2$ and $H^{-2}$ is denote by $\langle\cdot, \cdot\rangle$. We also use $C$ and $C_i$ to denote various positive constant that may have different values in different places.

Firstly, we introduce the Sobolev's embedding inequality : assume that $p$ be a constant such that $1\leq p\leq \frac{2n}{n-4}$ if $n\geq5$; $p\geq1$ if $n\leq4$, then $H_{0}^{2}(\Omega) \hookrightarrow L^{p}(\Omega)$ continuously, and

where $C_p$ denotes the best embedding constant.

Suppose (2) holds, we define

for any $\alpha\in [0,\alpha^*)$, then $H_{0}^{2}(\Omega) \hookrightarrow L^{p+\alpha}(\Omega)$ continuously. And we denote $C_{p+\alpha}$ by $C_*$.

Definition 2.1. A function $u \in C\left([0, T], H_0^{2}(\Omega)\right) \cap C^{1}\left([0, T], L^{2}(\Omega)\right) \cap C^{2}([0, T],\\ H^{-2}(\Omega))$ with $u_{t} \in L^{m}([0, T], L^m(\Omega))$ is called a weak solution to (1) if the following conditions hold

and

for any $v\in H_0^2(\Omega)$ and a.e. $t\in [0,T]$.

Now, we introduce the energy functional $J$ and the Nehari functional $I$ defined on $H_0^2(\Omega)\setminus \{0\}$ by

and

From the definitions (11) and (12), we have

The following lemmas play an important role in the studying the properties of the potential well.

Lemma 2.2. Let $u\in H_0^2(\Omega)\setminus \{0\}$ and $g(\lambda) = J(\lambda u)$. Then we have

(ⅰ): $\lim _{\lambda \rightarrow 0^{+}} J(\lambda u) = 0, \lim _{\lambda \rightarrow+\infty} J(\lambda u) = -\infty$;

(ⅱ): there exists a unique $\lambda^* = \lambda^*(u)>0$ such that $\frac{d}{d\lambda} J(\lambda u)\mid_{\lambda = \lambda^*} = 0$, and $J(\lambda u)$ is increasing on $(0, \lambda^*)$, decreasing on $(\lambda^*, +\infty)$ and taking the maximum at $\lambda^*$.

(ⅲ): $I(\lambda u)>0$ for $0<\lambda<\lambda^*, I(\lambda u)<0$ for $\lambda^*<\lambda<+\infty$ and $I(\lambda^* u) = 0$.

Proof. We know

It is obvious that $(i)$ holds due to $p\geq2$ and $\|u\|_p\neq0.$ Taking derivative of $g(\lambda)$ we obtain

and

From (14) and $p\geq2$, we see that there exists a unique positive $\lambda^{*}$ such that

then we obtain

Substituting the above equation into $g^{\prime \prime}(\lambda)$, we have

From these and $(i),$ we can yield that $g(\lambda)$ has a maximum value at $\lambda = \lambda^{*}$ and $J(\lambda u)$ increasing on $0 < \lambda \leq \lambda^{*}$ and decreasing on $\lambda^{*} \leq \lambda<+\infty$. So $(ii)$ holds.

From (12) and (14), we have

Then, we could define the potential well depth of the functional $J$ (also known as mountain pass level) by

We also define the well-known Nehari manifold

As in [29,34], that the mountain pass level $d$ defined in (15) can also be characterized as

It is easy to see that $d\geq0$ from (13). Now, we will prove that $d$ is strictly positive.

Lemma 2.3. Assume that $p>2$ holds. Let $\alpha\in (0, \alpha^*)$, and

Then, for any $u \in H_{0}^{2}(\Omega) \backslash\{0\},$ we have

(ⅰ) : if $\|\nabla u\|_{2} \leq r(\alpha),$ then $I(u)>0$

(ⅱ) : if $I(u) \leq 0,$ then $\|\Delta u\|_{2}>r(\alpha)$.

Proof. Since $\log y < y$ for any constant $y > 0$, using (9) and the definition of $I(u)$ in (12), we obtain that

Obviously, the results can be obtained from the above inequality (16).

Lemma 2.4. Assume the notations in Lemma 2.2 hold, we have

where $B = C_p$ as in (9) and $|\Omega|$ is the measure of $\Omega$.

Proof. It is obvious that $r_*>0$ (if exists), hence, we only need to prove $r(\alpha)\leq \gamma(\alpha)$, $r^*$ exists and $r^*<+\infty$, where

For any $u\in H_0^2(\Omega)$, using the Hölder's inequality, we have

Then, noticing $C_* = C_{p+\alpha}$ and $B = C_p$, we deduce

which implies

that is $r(\alpha)\leq \gamma(\alpha)$.

Now, we will prove $r^*$ exists and $r^*<+\infty$. For this purpose, we divide the proof into two cases.

Case a. If $n\geq5,$ we see that $\alpha \in\left(0, \alpha^{*}\right) = \left(0, \frac{2 n}{n-4}-p\right)$ and $\gamma(\alpha)$ is continuous on closed interval $[0, \frac{2 n}{n-4}-p]$. Hence, we have $r^*$ exists and

Case b. If $n\leq4,$ we define the following auxiliary function

Hence

For simplicity, we set

then

which yields that the function $g(\alpha)$ is strictly increasing on $\left(0, kB^{2}|\Omega|^{1-\frac{2}{p}}\right)$ and strictly decreasing on $\left(kB^{2}|\Omega|^{1-\frac{2}{p}},+\infty\right)$.

On the one hand, due to $p>2$, it is easy to see that

which implies that $g(\alpha)>0$ for $\alpha\in \left(0, kB^{2}|\Omega|^{1-\frac{2}{p}}\right)$ by $g(\alpha)$ is strictly increasing on $\left(0, kB^{2}|\Omega|^{1-\frac{2}{p}}\right)$.

While on the other hand, we can deduce that

which together with $g\left(kB^{2}|\Omega|^{1-\frac{2}{p}}\right)>0$ and $g(\alpha)$ is strictly decreasing on $\left(kB^{2}|\Omega|^{1-\frac{2}{p}},+\infty\right),$ implies that there exists a unique $\alpha_{*} \in\left(kB^{2}|\Omega|^{1-\frac{2}{p}},+\infty\right)$ such that $g\left(\alpha_{*}\right) = 0.$

Noting the relation between $h^{\prime}(\alpha)$ and $g(\alpha)$, we deduce that $h^{\prime}(\alpha)>0$ for $\alpha \in\left(0, \alpha_{*}\right),$ and $h^{\prime}(\alpha)<0$ for $\alpha \in (\alpha_*, +\infty)$. Therefore, $h(\alpha)$ achieves its maximum value at $\alpha = \alpha_{*}$, that is

Making using of the Lemmas 2.2 and 2.3, we obtain the following corollary.

Corollary 1. Assume that $p>2$ holds. Then, we have

(ⅰ): if $\|\nabla u\|_{2}<r_{*},$ then $I(u)>0$;

(ⅱ): if $I(u) \leq 0,$ then $\|\nabla u\|_{2} \geq r_{*}$

for any $u\in H_0^2(\Omega)\setminus \{0\}$, where $r_*$ is the positive constant defined in Lemma 2.3.

Lemma 2.5. Assume that $p\geq 2$ holds. Then the constant $d$ defined in (15) is strictly positive.

Proof. (ⅰ) For the case $p = 2$, we have $d\geq\frac{k}{4}\left(\frac{2 \pi}{k}\right)^{\frac{n}{2}} e^{n}$(see [9,20,25] for details).

(ⅱ) For the case $p>2$, by (ii) of Corollary 2.1, we get that $\|\nabla u\|_{2} \geq r_{*}$ if $u\in \mathcal{N}$. Then, making using of (13) with $I(u)>0$, we obtian

We define energy for the problem (1), which obeys the following energy equality of the weak solution $u$

where

It is obvious that

Taking $v = u_t$ in (10), after a simple calculation, we get

Now, we define the subsets of $H_{0}^{2}(\Omega)$ related to problem (1). Set

where $W$ and $V$ are called the stable and unstable set, respectively [21].

In order to establish the global existence and blow-up results of solution, we have to prove the following invariance sets of $W$ and $V$.

Lemma 2.6. If $u_{0} \in H_{0}^{2}, u_{1} \in L^{2}, p\geq 2, E(0)<d,$ and $u$ is a weak solution of problem (1) on $[0, T),$ where $T$ is the maximal existence time of weak solution, then

(ⅰ): $u \in W$ if $I\left(u_{0}\right)>0$;

(ⅱ): $u \in V \mathit{\text{if}} I\left(u_{0}\right)<0$.

Proof. It follows from the definition of weak solution and (17) that

(ⅰ) Arguing by contradiction, we assume that there exists a number $t_0 \in(0, T)$ such that $u(t) \in W$ on $\left[0, t_0\right)$ and $u\left(t_0\right) \notin W .$ Then, by the continuity of $J(u(t))$ and $I(u(t))$ with respect to $t,$ we have either (a)$J\left(u\left(t_{0}\right)\right) = d$ or (b) $I\left(u\left(t_{0}\right)\right) = 0$ and $\left\|u\left(t_{0}\right)\right\| \neq 0$.

It follows from (20) that (a) is impossible. If (b) holds, then by the definition of $d$, we have $J(u(t_0))\geq d$, which contradicts (20) again. Thus, we have $u(t) \in W$ for all $t \in[0, T)$.

(ⅱ) The proof is similar to the proof of (ⅰ). We omit it.

3.   Local existence

In this section, we are concerned with the local existence and uniqueness for the solution of the problem (1). The idea comes from [14,28,38], where the source term is polynomial. First, we give a technical lemma given in [22] which plays an important role in the uniqueness of the solution.

Lemma 3.1. ([22]) For every $\varepsilon>0,$ there exists $A>0,$ such that the real function

satisfies

Theorem 3.2. Suppose that $u_{0} \in H_0^{2}(\Omega), u_{1} \in L^{2}(\Omega),$ and $p>2$, then there is $T>0$, such that the problem (1) admits a unique local weak solution on $[0, T]$.

Proof. For every $T>0$, we consider the space

endowed with the norm

For every given $u\in \mathcal{H}$, we consider the following initial boundary value problem

We shall prove that the problem (21) admits a unique solution $v \in \mathcal{H} \cap C^{2}(|0, T|, H^{-2}(\Omega))$ with $v_{t} \in L^{m}(|0, T|, L^m( \Omega))$.

Let $W_{h} = \operatorname{Span}\left\{w_{1}, \ldots, w_{h}\right\}$, where $\left\{\omega_{i}\right\}_{i = 1}^{\infty}$ is the orthogonal complete system of eigenfunctions of $\Delta^2$ in $H_0^2(\Omega)$ with $\left\|w_{i}\right\| = 1$ for all $i$. Then, $\left\{w_{i}\right\}$ is orthogonal and complete in $L^{2}(\Omega)$ and in $H_{0}^{2}(\Omega) ;$ denote by $\left\{\lambda_{i}\right\}$ the related eigenvalues to their multiplicity. Let

such that $u_{0h} \in W_{h}, u_{1k}\in W_{h}, u_{0h} \rightarrow u_{0}$ in $H_{0}^{2}(\Omega)$ and $u_{1h} \rightarrow u_{1}$ in $L^{2}(\Omega)$ as $h \rightarrow \infty$. For each $h>0$ we seek $h$ functions $\gamma_{1 h}, \ldots, \gamma_{h h} \in C^{2}[0, T]$ such that

solves the following problem

For $i = 1, \ldots, h,$ taking $\eta = w_{i}$ in (23) yields the following Cauchy problem for a nonlinear ordinary differential equation with unknown $\gamma_{i h}$

where

Then the above problem admits a unique local solution $\gamma_{ih}\in C^{2}[0, T]$ for all $i,$ which in turn implies a unique $v_{h}$ defined by (22) satisfying (23).

Taking $\eta = v_{h}^{\prime}(t)$ into (23) and integrating over $[0, t]\subset [0, T],$ we have

for every $h\geq1$. We estimate the last term in the right-hand side of (24) thanks to Young's and Sobolev's inequalities

In order to estimate (25), we focus on the logarithmic term. Here we denote $\Omega : = \Omega_{1} \cup \Omega_{2},$ where $\Omega_{1} = \{x \in \Omega| | u(x) |<1\}$ and $\Omega_{2} = \{x \in \Omega| | u(x) | \geq 1\}.$ Then we have

By a simple calculation, we obtain

which implies

Let

By the Sobolev embedding from $H_0^2(\Omega)$ to $L^{\frac{2 n}{n-4}}(\Omega)$ if $n\geq 5$ and to $L^q(\Omega)$ for any $q\geq1$ if $n\leq4$, recalling $u \in \mathcal{H} : = C([0, T] ; H_0^2(\Omega))$, we have

The proof of the case $n\leq4$ is similar. From the above discussion, (25) yields

Substituting this inequality into (24), we obtain

where $C>0$ is independent of $h$. It follows from (27) that

Hence, up to a subsequence, we could pass to the limit in (23) and obtain a weak solution $v$ of (21) with regularity (28). Then, we have $v \in C\left([0, T], H_{0}^{2}(\Omega)\right)\cap C^{1}\left([0, T], L^{2}(\Omega)\right)$ with $v_{t} \in L^{m}([0, T], L^m(\Omega))$. Finally, from (21), we obtain $v^{\prime \prime} \in C^{0}([0, T], H^{-2}(\Omega))$. Then the weak local solution of problem (21) has been obtained.

To prove the uniqueness, arguing by contradiction: if $w$ and $v$ were two solutions of (21) which have the same initial data. Subtracting these two equations and testing the result by $w_{t}-v_{t},$ we could obtain

It follows from the following element inequality

that (29) can make to be

Therefore, we have $w = v$, i.e. the problem (21) obeys a unique weak solution.

Now, we are in the position to prove Theorem 3.1. For $u_{0} \in H_{0}^{2}(\Omega), u_{1} \in L^{2}(\Omega),$ we denote

and

for every $T>0$. From the above discussion, for any $u\in B_{RT}$, we could introduce a map $\Phi : \mathcal{H} \rightarrow \mathcal{H}$ defined by $v = \Phi(u),$ where $v$ is the unique solution to (21).

Claim. $\Phi$ is a contract map satisfying $\Phi\left(B_{RT}\right) \subseteq B_{RT},$ for small $T>0$.

In fact, assume that $u \in B_{RT},$ the corresponding solution $v = \Phi(u)$ satisfies (21) for all $t \in[0, T]$. Thus, as did the proof (26) and (27), we have

for $n\geq5$, For the case $n\leq4$, the index $\frac{2n}{n-4}$ of $R$ in the last inequality can be replaced by any fixed positive number. If $T$ is sufficiently small, then $\|v\|_{\mathcal{H}} \leq R,$ which implies that $\Phi\left(B_{RT}\right) \subseteq B_{RT}$

Next we show that $\Phi$ is contractive in $B_{RT}$. We set $v_{1} = \Phi\left(w_{1}\right), v_{2} = \Phi\left(w_{2}\right)$ with $w_{1}, w_{2} \in B_{RT},$ and $v = v_1-v_2$, then, $v$ satisfies

for any $\eta\in H_0^2(\Omega)$ and a.e. $t \in[0, T]$.

Taking $\eta = v_{t} = v_{1 t}-v_{2 t}$, noticing

and integrating both sides of (30) over $(0, t)$, we have

We need estimating the logarithmic term in (31) by using Lemma 3.1. By the similar argument as [22], we give the sketch of the proof.

Making use of mean value theorem, we have, for $0 < \theta< 1$,

Then, it follows from Lemma 3.1 that

Since $w_1, w_2 \in B_{RT}$, using Hölder's inequality and the Sobolev embedding, we can obtain

By the similar argument, we have

Using (2), we can choose sufficiently small $\varepsilon>0$ such that

which yields that

Noticing $\|w_1-w_2\|\leq C\|w_1-w_2\|_{H_0^2(\Omega)},$ from the above discussions, we can deduce that

Thus, it follows from (31) that

We choose $T$ sufficiently small such that $C\left(R^{p-2}+1+R^{\bar{p}(p-2)/2(p-1)}\right)T <1$. Thus from (32) we obtain $\Phi$ is a contract map in $B_{RT}.$ The contraction mapping principle then shows that there exists a unique $u\in B_{RT}$ satisfying $u = \Phi(u)$ which is a solution to problem (1). The proof is complete.

4.   Global existence and energy decay

In this section, we consider the global existence and energy decay of the solution for problem (1). First, we introduce the following lemmas which play an important role in studying the decay estimate of global solution for the problem (1).

Lemma 4.1. [33] Let $\phi(t)$ be a nonincreasing and nonnegative function on $[0, T],$ $T>1,$ such that

where $\omega_{0}$ is a positive constant and $r$ is a nonnegative constant. Then we have

(ⅰ): if $r>0,$ then

(ⅱ): if $r = 0,$ then

where $\omega_{1} = \log \left(\frac{w_{0}}{\omega_{0}-1}\right),$ here $\omega_{0}>1$.

Now, we establish the global existence and energy decay results.

Theorem 4.2. Let $u$ be the unique local solution to problem (1). Assume (2) and $2\leq m<p$ hold. If $u_0\in W$, $u_1\in L^2(\Omega)$ and $E(0)<d$, then $u(t)$ is the global solution to the problem (1). Moreover it has the following decay property

and

where $K$ and $\kappa$ are positive constants, $\tau$ is given by (47).

Proof. Step 1.. Global existence. It suffices to show that $\left\|u_{t}\right\|^{2}+\|\Delta u\|^{2}$ is uniformly bounded with respect to $t$. It follows from Lemma 2.5 (i) that $u(t)\in W$ on $[0, T]$. Using (13), we have the following estimate

which yields that

The above inequality and the continuation principle imply the global existence, i.e. $T = +\infty$.

Step 2.. We claim that there exists constant $\theta \in (0,1)$ such that

In fact, it follows from $I(u(t))>0$ for all $t\geq 0$ and Lemma 2.1 that there exists a $\lambda_0>1$ such that $I(\lambda_0u(t)) = 0$. Making use of (33), we have

which implies that

It follows from (12) that

Combining this equality with (35), we have

which implies that

Hence, the inequality (34) holds with $\theta = 1-\lambda_0^{2-p}$.

Step 3.. Energy decay. By integrating (18) over $[t, t+1], t>0,$ we obtain

where

In view of (37) and the embedding $L^{m}(\Omega)\hookrightarrow L^{2}(\Omega)$, we obtain

Thus, from (38), there exist $t_{1} \in\left[t, t+\frac{1}{4}\right]$ and $t_{2} \in\left[t+\frac{3}{4}, t+1\right]$ such that

On the other hand, multiplying $(1.1)_1$ by $u$ and integrating over $\Omega \times\left[t_{1}, t_{2}\right],$ we have

It follows from (33) that

By using (33) and (39), we also have

Combining (38), (41) with (42), we have from (40) that

Moreover, using (33) and (34), it is easy to see that

Thus, we deduce that

where $C_2 = \frac{1}{p}+\frac{p-2}{2p\theta}+\frac{kC_p^p}{p^2\theta}\left(\frac{2p}{p-2}E(0)\right)^{\frac{p-2}{2}}$. By integrating (44) over $\left(t_{1}, t_{2}\right),$ we have

By integrating (18) over $[t, t_2]$, we obtain

Since $t_{2}-t_{1} \geq \frac{1}{2},$ it easy to see that

Then, in view of (36), we have

Thus, combining (38) with (45), we get that

Hence, it follows from Young's inequality that

holds with some positive constant $C_3>1$. It follows from (36) and (46), we deduce that

which implies that

where $C_{4}(E(0)) = C_{3}\left[1+E(0)^{\frac{m-2}{m}}+E(0)^{\frac{2 m-4}{m}}\right]$. Notice that

Hence, the energy decay estimates hold with

5.   Blow up for negative energy

In this section, we will establish that the solution of problem (1) blows up in finite time provided $E(0)<0.$ For this purpose, we give some useful lemmas.

Lemma 5.1. Assume that (2) holds. Then there exists a positive constant $C$ such that

for any $u\in H_0^2(\Omega)$ and $2\leq s\leq p$, provided that $\int_{\Omega}|u|^{p} \log |u|^{k} \mathrm{d} x \geq 0$.

Lemma 5.2. Assume that (2) holds. Then there exists a positive constant $C$ such that

for any $u\in H_0^2(\Omega)$, provided that $\int_{\Omega}|u|^{p} \log |u|^{k} \mathrm{d} x \geq 0$.

Lemma 5.3. Assume that (2) holds. Then there exists a positive constant $C>1$ such that

for any $u\in H_0^2(\Omega)$ and $2\leq s\leq p$.

The proof of lemma 5.1-5.3 is similar to the proof in [22], we omit the details.

Lemma 5.4. Assume that (2) and $m<p$ hold. Then there exists a positive constant $C$ such that

for any $u\in H_0^2(\Omega)$, provided that $\int_{\Omega}|u|^{p} \log |u|^{k} \mathrm{d} x \geq 0$.

Proof. Noting $m\leq p$ and using the fact that $\|u\|_m^m\leq C\left(\|u\|_p^p\right)^{\frac{m}{p}}$, we can obtain the result from Lemma 5.2.

Now we are in the position to state and prove the blow up result for $E(0)<0$.

Theorem 5.5. Suppose that the conditions in Lemma 5.4 hold. Then the solution to the problem (1) blows up in finite time provided that $E(0)<0$.

Proof. We denote $H(t) = -E(t)$. It follows from (17) and (18) that

and

We define

where $\varepsilon>0$ to be determined later and

By taking a derivation of $L(t)$, we get

Adding and subtracting $\varepsilon p(1-a)H(t)$ for some $a\in(0,1)$ in the RHS of the above equation, then using the definition of $H(t)$, we obtain

In view of Young's inequality, we have

for any $\delta>0$, which yields, by substitution in (50),

Since the integral is taken over the $x$ variable, (50) holds even if $\delta$ is time dependent. Thus by choosing $\delta$ so that $\delta^{-m /(m-1)} = M H^{-\beta}(t)$, for large $M$ to be determined later, substituting in (51), we obtain

Making using of (48), Lemma 5.4 and Young's inequality, we find

Hence, it follows from Lemma 5.1 that

Thus, Lemma 5.1 implies

Combining (52) and (53), we have

Now, we choose $a>0$ sufficiently small that

and $M$ sufficiently large that

Once $M$ and $a$ are fixed, we choose $\varepsilon$ sufficiently small that

Thus, for some constant $\gamma>0$, (54) has the form

Consequently we have

On the other hand, using Lemma 5.3, by the same method as in [32], we can deduce

Combining (55) and (56), we obtain

where $\lambda>0$ is constant depending only on $\gamma$ and $C$. By a simple integration of (57) over $(0, t)$, we have

which implies that $L(t)$ blow up in finite time

This completes the proof of Theorem 5.1.

6.   Arbitrarily high initial energy for linear damping

In this section, we consider the problem (1) with the linear damping term, i.e. $m = 2$. We will establish the finite time blow-up result by the method of the so called concavity method. For simplicity, we denote $\|u\|^2\leq B_0\|\Delta u\|^2.$

Lemma 6.1. [14] Let $\delta\geq0, T>0$ and $h$ be a Lipschitzian function over $[0, T)$. Assume that $h(0)\geq0$ and $h'(t)+\delta h(t)>0$ for a.e. $t\in [0, T)$. Then $h(t)>0$ for all $t\in(0,T).$

Lemma 6.2. Suppose that $m = 2$. Let $I(u_0)<0$ and $u_1\in L^2(\Omega)$ such that

Let $u(t)$ be the solution with the initial data $(u_0, u_1)$. Then the map $\{t\rightarrow \|u(t)\|^2\}$ is strictly increasing provided $I(u(t))<0$.

Proof. Let $F(t) = \|u(t)\|^2$ and $G(t) = F'(t) = 2\int_{\Omega}uu_t\mathrm{d}x$. A direct computation yields

Moreover, by testing the equation with $u(t)$, we have

which implies

Hence, if $I(u(t))>0$, we can deduce

Therefore, it follows from Lemma 6.1 with $\delta = 1$ that $G(t) = F'(t)>0$. Hence $F(t)$ is strictly increasing provided $I(u(t))<0$.

Lemma 6.3. Let $m = 2$. Assume that $u_0\in H_0^2(\Omega)$, $u_1\in L^2(\Omega)$ and (2) holds. Assume that the initial data satisfies

where $\Lambda = \frac{2B_0p}{p-2}$. Then the solution $u(t)$ of the problem (1) with $E(0)>0$ satisfies $I(u(t))<0$ provided $I(u_0)<0$.

Proof. If this was not the case, by the continuity of $I(u(t))$ in $t$, then there would exist a first time $t_0\in (0, T)$ such that $I(u(t_0)) = 0$ and $I(u(t))<0$ for $t\in[0,t_0)$. It follows form the Cauchy-Schwarz inequality that

By Lemma 6.2, (58) and (59), we deduce that

which implies

by the continuity of $u(t)$ in $t$. Moreover, it follows from (13) and (17) that

that is

Hence, we have

which is a contradiction with (61). The proof is complete.

We now present the main blow-up result for the weak solution of problem (1) with $m = 2$ with arbitrary positive initial energy.

Theorem 6.4. Assume the conditions of Lemma 6.3 hold. Then the weak solution $u(t)$ of the problem (1) blows up in finite time provided that $E(0)>0$ and $I(u_0)<0$.

Proof. It follows from Lemma 6.3 that $I(u(t))<0$ for $t\in [0, T)$. By contradiction, we assume now that $u(t)$ is global, namely $T = \infty$. Then, for any $T_0>0$, we may consider $\eta: [0,T_0]\rightarrow \mathbb{R}^+$ defined by

Notice $\eta(t)>0$ for all $t\in [0, T_0]$; hence, since $\eta$ is continuous, there exists $\varrho>0$ (independent of the choice of $T_0$) such that

Moreover,

hence, we have

It follows from (63) that

By the Cauchy-Schwarz inequality, we obtain

and

Combining the above inequalities, we have

Hence, it follows from (64) and (65) that

Now, we define

Noticing $m = 2$ in this section, using (13) and (17), we obtain

From (60) and Lemma 6.2, we deduce that

which yields that $\xi(t)\geq \varsigma>0$. Then, (66) can be rewritten as

which implies that

Hence, it follows that there exists a $T^*>0$ such that

that is

In turn, this implies that

In fact, if $\|u(t)\|\rightarrow+\infty$ as $t\rightarrow T^*$, then (67) immediately follows. On the contrary, if $\|u(t)\|$ remains bounded on $[0, T^*)$, then

so that (67) is also satisfied. Hence (67) is a contraction with $T = +\infty$. The proof is complete.

Acknowledgments

The author would like to thank Professor Hua Chen for the careful reading of this paper and for the valuable suggestions to improve the presentation of the paper. This project is supported by NSFC (No. 11801145), Key Scientific Research Foundation of the Higher Education Institutions of Henan Province, China (Grant No.19A110004 and the Fund of Young Backbone Teacher in Henan Province (NO. 2018GGJS068, 21420048).