On a logarithmic wave equation with nonlinear dynamical boundary conditions: local existence and blow-up

Abstract

This paper deals with a hyperbolic-type equation with a logarithmic source term and dynamic boundary condition. Given convenient initial data, we obtained the local existence of a weak solution. Local existence results of solutions are obtained using the Faedo-Galerkin method and the Schauder fixed-point theorem. Additionally, under suitable assumptions on initial data, the lower bound time of the blow-up result is investigated.

1 Introduction

In this paper, we study the problem of wave equation with logarithmic nonlinearity and dynamic boundary condition

$$\textstyle\begin{cases} u_{tt}-\triangle u=0, & \text{in }(0,\infty )\times \Omega, \\ u(x,t)=0 & \text{on }[0,\infty )\times \Gamma _{0}, \\ \frac{\partial }{\partial n}u(x,t)=- \vert u_{t} \vert ^{k-2}u_{t}+|u|^{p-2}u \ln |u| & \text{on }[0,\infty )\times \Gamma _{1}, \\ u(x,0)=u_{0}(x),\qquad u_{t}(0,x)=u_{1}(x) & \text{on }\Omega ,\end{cases}$$
(1)

where $$\Omega \subset R^{n}$$, $$n \geq 1$$ is a regular, bounded domain with a boundary $$\partial \Omega =\Gamma _{0} \cup \Gamma _{1}$$, $$\Gamma _{0} \cap \Gamma _{1}=$$, where $$\Gamma _{0}$$ and $$\Gamma _{1}$$ are measurable over Ω, endowed with the $$(n-1)$$ dimensional Lebesgue measures $$\lambda _{n-1} (\Gamma _{0} )$$ and $$\lambda _{n-1} (\Gamma _{1} )$$. Additionally, $$\lambda _{n-1} (\Gamma _{0} )$$ and $$\lambda _{n-1} (\Gamma _{1} )$$ are assumed to be positive throughout paper. $$k \geq 2$$ and $$p \geq 2$$ are positive constants to be chosen later.

Dynamic boundary problems are widely applied in many mathematical models, such hydro logic filtration process, thermoelasticity, diffusion phenomenon, and hydrodynamics [2, 15, 2527]. A dynamic boundary condition has been introduced by a group of physicists to underline the fact that the kinetics of the process, i.e. the term $$\frac{\partial u}{\partial n}$$ becomes more visible in some boundary conditions [18, 24]. This type of option is characterized by the interaction of the components of the system with the walls (i.e., within Γ) [7]. Since the paper by Lions [29] has been introduced in 1969, evolution equations with dynamical boundary conditions (first order equations in time) have been studied well. Later, mathematicians and physicists studied it for a long time and achieved creative success; see [3, 6, 9, 17, 19, 20, 22, 28, 31] and references therein.

In [31], the author considered the problem (1) without logarithmic source term for $$\frac{\partial}{\partial n} u(x, t)= - \vert u_{t} \vert ^{k-2} u_{t}+|u|^{p-2} u$$ boundary condition and proved the local and global existence under suitable condition. When $$2< p< k$$, the solutions exist globally for arbitrary initial data. For $$k< p$$, solutions blow up. Later, Zhang and Hu [36] considered the blow-up of the solution under the condition $$E(0)< d$$ when the initial data are in the unstable set. In [12], they established blow-up results of the solution for a finite time at a critical energy level or high-energy level for the same problem.

Let us go back and look at a wave equation with logarithmic nonlinearity associated with problem (1). In [8], Cazenave and Haraux considered the following equation for the Cauchy problem

$$u_{t t}-\triangle u=k u \ln |u| \text{. }$$
(2)

They studied deeply the existence and uniqueness of the solutions using different techniques. As far as is known, this type of problem has been employed in various areas of physics, such as geophysics, nuclear physics, and optics; see in Bialynicki-Birula and Mycielski [4, 5]. Moreover, there are many research points devoted to the given problem in different models of hyperbolic wave equation with logarithmic source term [10, 13, 14, 16, 21, 23, 33]. Ma and Fang [32] considered problem (2) with strong damping term. They proved decay estimates and blow-up result under the null Dirichlet boundary condition.

In [11], Cui and Chai considered the following equation

$$u_{t t}-{\operatorname{div}}\bigl(A(x) \nabla u\bigr)=|u|^{p} u \ln |u|$$

with acoustic boundary condition. They obtained local existence and uniqueness using the semigroup theory. As far as is known, not many works are related to the logarithmic wave equation with a dynamic boundary condition. According to the studies mentioned above, our work aims to expand the result of wave equation with logarithmic nonlinearity and dynamic boundary conditions. The rest of the work is arranged as follows: In Sect. 2 gives notations and lemmas to illustrate our paper path. Sections 34 state the local existence result and potential well of (1). In the last part, we established blow-up result for a lower bound time.

2 Preliminaries

First, we denote

$$\Vert \cdot \Vert =L^{2}(\Omega ),\qquad \Vert \cdot \Vert _{q}=L^{q}( \Omega ),\qquad \Vert \cdot \Vert _{q,\Gamma _{1}}=L^{q} ( \Gamma _{1} ) ,\quad 1\leq q\leq \infty$$

and

$$H_{\Gamma _{0}}^{1}(\Omega )= \bigl\{ u\in H^{1}( \Omega ): u | _{\Gamma _{0}}=0 \bigr\} ,$$

($$u|_{\Gamma _{0}}$$ is in the trace sense). Let $$T>0$$ be a real number and X be a Banach space endowed with norm $$\|\cdot \|_{x} . L^{p}(0, T ; X)$$ indicates the space of functions h, which are $$L^{p}$$ over $$(0, T)$$ with values in X, which are measurable with $$\|h\|_{x} \in L^{p}(0, T)$$. We set the Banach space endowed with the norm

$$\Vert h\Vert _{L^{p}(0,T;X)}= \biggl( \int _{0}^{T} \Vert h \Vert _{X}^{p} \biggr) ^{\frac{1}{p}}.$$

$$L^{\infty}(0, T ; X)$$ denotes the space of functions $$h:(0, T) \rightarrow X$$, which are measurable with $$\|h\|_{x} \in$$ $$L^{\infty}(0, T)$$. We set the Banach space endowed with the norm

$$\Vert h \Vert _{L^{p} ( 0,T;X ) }= \biggl( \int _{0}^{T} \Vert h \Vert _{X}^{p} \biggr) ^{\frac{1}{p}}.$$

We know that if X and Y are Banach spaces such that X is continuous embedding to Y, then $$L^{p}(0, T ; X) \hookrightarrow L^{p}(0, T ; Y)$$ for $$1 \leq p \leq \infty$$.

We define the total energy function as

$$E(t)=\frac{1}{2} \Vert u_{t} \Vert ^{2}+ \frac{1}{2}\Vert \nabla u\Vert ^{2}+\frac{1}{p^{2}} \Vert u\Vert _{p,\Gamma _{1}}^{p}- \frac{1}{p}\int _{\Gamma _{1}}|u|^{p}\ln |u|\,dx .$$
(3)

By the definition of $$E(t)$$ on $$H_{\Gamma _{1}}^{1}(\Omega )$$, the initial energy can be considered

$$E(0)=\frac{1}{2} \Vert u_{1} \Vert ^{2}+ \frac{1}{2} \Vert \nabla u_{0} \Vert ^{2}+ \frac{1}{p^{2}} \Vert u_{0} \Vert _{p,\Gamma _{1}}^{p}- \frac{1}{p} \int _{\Gamma _{1}} \vert u_{0} \vert ^{p}\ln \vert u_{0} \vert \,dx .$$
(4)

Lemma 1

[1] (Trace-Sobolev Embedding inequality). Let $$H_{\Gamma _{0}}^{1}(\Omega )\hookrightarrow L^{p} ( \Gamma _{1} )$$ for $$2\leq p<\varkappa$$ hold, where

$$\varkappa =\textstyle\begin{cases} \frac{2(n-1)}{n-2}, & \textit{if }n\geq 3, \\ \infty , & \textit{if }n=1,2.\end{cases}$$

So that, there is a constant $$C_{p}$$ that is the smallest nonnegative number, satisfying

$$\Vert u\Vert _{p,\Gamma _{1}}\leq C_{p} \Vert \nabla u_{0} \Vert .$$
(5)

Proposition 2

Suppose that Lemma 1holds, we define

$$\alpha ^{\ast }=\textstyle\begin{cases} \frac{2(n-1)}{n-2}-p,&\textit{if }n\geq 3, \\ \infty ,& \textit{if }n=1,2\end{cases}$$

for any $$\alpha \in [0, \alpha ^{*} )$$, then $$H_{\Gamma _{0}}^{1}(\Omega ) \hookrightarrow L^{p+\alpha} (\Gamma _{1} )$$ continuously.

Lemma 3

$$E(t)$$ is a nonincreasing function for $$0\leq s\leq t\leq T$$ and

$$E(t)+ \int _{s}^{t} \bigl\Vert u_{\tau }(\tau ) \bigr\Vert _{k, \Gamma _{1}}^{k}\,d\tau =E(s)\leq 0.$$
(6)

Proof

By multiplying equation (1) by $$u_{t}$$ and integrating on Ω, we have

\begin{aligned}& \int _{\Omega} u_{t t} u_{t} \,dx - \int _{\Omega} \Delta u u_{t} \,dx =0, \\& \frac{d}{d t} \biggl(\frac{1}{2} \Vert u_{t} \Vert ^{2}+\frac{1}{2}\| \nabla u\|^{2} \bigg|_{s} ^{t}+\frac{1}{p^{2}}\|u \|_{p, \Gamma _{1}}^{p}- \frac{1}{p} \int _{\Gamma _{1}}|u|^{p} \ln |u| \,dx \biggr)=- \Vert u_{t} \Vert _{k, \Gamma _{1}}^{k} . \end{aligned}
(7)

By integrating of (7) over $$(s, t)$$, we have equality (6). □

Lemma 4

Let ϑ be a positive number. Then, the inequality holds

$$\bigl\vert \vert s \vert ^{p-2}\log \vert s \vert \bigr\vert \leq A+s^{p-2+\vartheta },\quad p>2$$
(8)

for $$A>0$$.

Proof

Notice that $$\lim_{|s|\rightarrow \infty }\frac{\ln |s|}{s^{\vartheta }}=0$$. Then, there is a positive constant $$K>0$$ such that

$$\frac{\log |s|}{s^{\vartheta }}< 1$$

for $$\forall |s|>K$$. Therefore,

\begin{aligned} &\log |s| < s^{\vartheta} \\ &|s|^{p-2} \log |s| < s^{p-2+\vartheta}, \end{aligned}

for $$\forall |s|>K$$. Since $$p>2$$, then $$|| s |^{p-2} \log |s| |\leq A$$, for some $$A>0$$ and for all $$|s| \leq K$$.

Thus,

$$\bigl\vert \vert s \vert ^{p-2}\log |s| \bigr\vert \leq A+s^{p-2+\vartheta },\quad p>2.$$

□

3 Existence of local solution

We will apply the Faedo-Galerkin technique and the Schauder fixed-point theorem.

Theorem 5

There exists $$T>0$$, such that problem (1) has a unique local weak solution u of (1) on $$(0,T)\times \Omega$$. Therefore,

$$\begin{gathered} u \in C \bigl([0, T] ; H_{\Gamma _{0}}^{1}( \Omega ) \bigr) \cap C^{1} \bigl([0, T\bigr) ; L^{2}( \Omega ) ), \\ u_{t} \in L^{k} \bigl((0, T) \times \Gamma _{1} \bigr) \end{gathered}$$

and the energy identity

$$\frac{1}{2} \Vert u_{t} \Vert ^{2}+ \frac{1}{2} \Vert \nabla u\Vert ^{2}\bigg| _{s}^{t}+{ \int}_{s}^{t} \Vert u_{t} \Vert _{k,\Gamma _{1}}^{k}={ \int}_{s}^{t} \int _{ \Gamma _{1}}|u|^{p}\ln |u|u_{t}\,dx$$

holds for $$0 \leq s \leq t \leq T$$. Therefore, $$T=T ( \Vert u_{0}\|_{H_{\Gamma _{0}}^{1}(\Omega )}^{2}+ \Vert u_{1} \Vert ^{2} |_{s} ^{t}, k, p, \Omega , \Gamma _{1} )$$ is decreasing in the first variable.

Now, we will give some existence result and lemma used for the proof of Theorem 5.

To define the function and show that the fixed point exists, we introduce the following problem:

$$\textstyle\begin{cases} v_{tt}-\triangle v=0, & \text{in }(0,T)\times \Omega, \\ v=0 & \text{on }[0,T)\times \Gamma _{0}, \\ \frac{\partial }{\partial n}v(x,t)=- \vert v_{t} \vert ^{k-2}v_{t}+|u|^{p-2}u \ln |u| & \text{on }[0,T)\times \Gamma _{1}, \\ v(x)=u_{0}(x),v_{t}(x)=u_{1}(x) & \text{on }\Omega .\end{cases}$$
(9)

Let the solution v of problem (9) be $$v=\zeta (u)$$. We can see that v corresponds to u and $$\zeta : X_{T} \rightarrow X_{T}$$.

Lemma 6

Let $$2\leq p\leq \varkappa$$ and $$\frac{\varkappa }{\varkappa -p+1}< k$$. Assume that $$u\in H_{\Gamma _{0}}^{1}(\Omega )$$ and $$u_{1}\in L^{k}(\Omega )$$ hold. Then, there exists a unique weak solution u of (9) on $$(0,T)\times \Omega$$. Therefore,

$$Y_{T}= \bigl\{ ( v,v_{t} ) \in C \bigl( [0,T];H_{\Gamma _{0}}^{1}( \Omega ) \bigr) \cap C^{1} \bigl( [0,T\bigr);L^{2}(\Omega ) ) ,v_{t} \in L^{k} \bigl( (0,T)\times \Gamma _{1} \bigr) \bigr\}$$
(10)

endowed with the norm

$$\bigl\Vert ( v,v_{t} ) \bigr\Vert _{Y_{T}}^{2}= \max_{0 \leq t\leq T} \biggl[ \Vert v_{t} \Vert ^{2}+\frac{1}{2} \Vert \nabla v\Vert ^{2} \biggr] + \Vert v_{t} \Vert _{L^{k} ( (0,T)\times \Gamma _{1} ) }^{k},$$
(11)

and the energy identity

$$\frac{1}{2} \Vert v_{t} \Vert ^{2}+ \frac{1}{2} \Vert \nabla v\Vert ^{2}\bigg| _{s}^{t}+{ \int}_{s}^{t} \Vert v_{t} \Vert _{k,\Gamma _{1}}^{k}={ \int}_{s}^{t} \int _{ \Gamma _{1}}|u|^{p}\ln |u|u_{t}\,dx ,$$
(12)

holds for $$0 \leq s \leq t \leq T$$.

To see the first step of the proof of Lemma 6, we will use the following proposition. The proposition was proved similar to [35]. We have some results in [35] as follows:

Proposition 7

Let $$2\leq p\leq \varkappa$$ and $$\frac{\varkappa }{\varkappa -p+1}< k$$. Assume that $$u\in H_{\Gamma _{0}}^{1}(\Omega )$$ and $$u_{1}\in L^{k}(\Omega )$$ hold. Then, there is $$T>0$$ and a unique solution v for (9) problem on $$(0,T)$$ such that, i.e.

$$u\in L^{\infty } \bigl( [0,T];H_{\Gamma _{0}}^{1}(\Omega ) \bigr)$$

such that

$$u_{t}\in L^{\infty } \bigl( 0,T;L^{2}(\Omega ) \cap L^{k} \bigl( (0,T) \times \Gamma _{1} \bigr) \bigr)$$

and

$${ \int}_{0}^{T}\Omega -u_{t}\varphi _{t}+\nabla u\nabla \varphi +{ \int}_{0}^{T}\Gamma _{1} \vert u_{t} \vert ^{k-2}u_{t}\varphi -{ \int}_{0}^{T}\Gamma _{1}|u|^{p-2}u\ln |u|\varphi =0$$

for all $$\varphi \in C ((0, T) ; H_{\Gamma _{0}}^{1}(\Omega ) ) \cap C^{1} ((0, T) ; L^{2}(\Omega ) ) \cap L^{k} ((0, T) \times \Gamma _{1} )$$. Then

$$u\in C \bigl( [0,T];H_{\Gamma _{0}}^{1}(\Omega ) \bigr) \cap C^{1} \bigl( [0,T];L^{2}(\Omega ) \bigr) ,$$

and the energy identity

$$\frac{1}{2} \Vert u_{t} \Vert ^{2}+ \frac{1}{2} \Vert \nabla u\Vert ^{2}\bigg| _{s}^{t}+{ \int}_{s}^{t} \Vert u_{t} \Vert _{k,\Gamma _{1}}^{k}={ \int}_{s}^{t} \int _{ \Gamma _{1}}|u|^{p}\ln |u|u_{t}\,dx$$

holds for $$0 \leq s \leq t \leq T$$. Now, we can state the proof of Lemma 6.

Proof

Let $$\{ w_{j} \} _{j=1}^{\infty }$$ be a sequence of linearly independent vectors in $$X= \{ u\in H_{\Gamma _{0}}^{1}(\Omega ): u |_{ \Gamma _{1}}\in L^{k} ( \Gamma _{1} ) \}$$ whose finite linear combinations are dense in X. In the event, using the Grahm-Schmidt orthogonalization method, we can conclude $$\{ w_{j} \} _{j=1}^{\infty }$$ to be orthonormal in $$L^{2}(\Omega )\cap L^{2} ( \Gamma _{1} )$$. Using some technical mathematical result, we can clearly see that $$X ( u\in H_{\Gamma _{0}}^{1}(\Omega )\cap L^{k} ( \Gamma _{1} ) )$$ is dense in $$H_{\Gamma _{0}}^{1}(\Omega )$$ and in $$L^{2}(\Omega )$$. Moreover, there exist $$u_{0m},u_{1m}\in [ w_{1},w_{2},\ldots, w_{m} ]$$ where $$w_{1},w_{2},\ldots, w_{m}$$ are the span of the vectors such that

$$\begin{gathered} u_{0m}=\sum _{i=1}^{m} \biggl( \int _{\Omega }u_{0}w_{i} \biggr) w_{i} \rightarrow u_{0}\quad \text{in }H_{\Gamma _{0}}^{1}( \Omega ), \\ u_{1m}=\sum_{i=1}^{m} \biggl( \int _{\Omega }u_{1}w_{i} \biggr) w_{i} \rightarrow u_{1}\quad \text{in }L^{2}( \Omega ).\end{gathered}$$
(13)

According to their multiplicity of

$$\Delta w_{i}+\lambda _{i}w_{i}=0$$

we denote by $$\{\lambda _{i} \}$$ the related eigenvalues to $$w_{1}, w_{2}, \ldots, w_{m}$$. For all $$m \geq 1$$, we will seek an approximate solution (m functions $$\gamma _{i m}$$) such that

$$v_{m}(t)=\sum_{i=1}^{m} \gamma _{i}^{m}(t)w_{i}$$
(14)

satisfying the following Cauchy problem

$$\left \{ ( v_{mtt},w_{i} ) + ( \nabla v_{m},\nabla w_{i} ) + \int _{\Gamma _{1}} \vert v_{mt} \vert ^{k-2}v_{mt}w_{i}= \int _{\Gamma _{1}}|u|^{p-2}u\ln |u|w_{i}, \right .$$
(15)

where $$t \geq 0$$. In (15), for the first term, we obtain

$$\int _{\Omega }v_{mtt}(t)w_{i}\,dx = \int _{\Omega } \Biggl( \sum_{j=1}^{m} \ddot{\gamma}_{jtt}^{m}(t)w_{j} \Biggr) w_{i}\,dx =\ddot{\gamma}_{i}^{m}(t) \int _{\Omega } \vert w_{i} \vert ^{2}\,dx =\ddot{\gamma}_{i}^{m}(t).$$
(16)

Similarly,

\begin{aligned} \int _{\Omega}-\Delta v_{m} w_{i} \,dx & =- \int _{\Omega} \Delta \Biggl(\sum_{j=1}^{m} \gamma _{j}^{m}(t) w_{j} \Biggr) w_{i} \,dx \\ & =- \int _{\Omega} \Biggl(\sum_{j=1}^{m} \gamma _{j}^{m}(t) \Delta w_{j} \Biggr) w_{i} \,dx \\ & = \int _{\Omega} \sum_{j=1}^{m} \gamma _{j}^{m}(t) \lambda _{j} w_{j} w_{i} \,dx \\ & =\gamma _{i}^{m}(t) \lambda _{i} \int _{\Omega} \vert w_{i} \vert ^{2} \,dx \\ & =\gamma _{i}^{m}(t) \lambda _{i} . \end{aligned}
(17)

For the fourth term, we get

\begin{aligned} \int _{\Gamma _{1}} \bigl\vert v_{m t}(t) \bigr\vert ^{k-2} v_{m t}(t) w_{i} \,dx & = \int _{\Gamma _{1}} \Biggl(\sum_{j=1}^{m} \bigl\vert \dot{\gamma}_{j}^{m}(t) \bigr\vert ^{k-2} \dot{\gamma}_{j}^{m}(t) w_{j} \Biggr) w_{i} \,dx \\ & = \bigl\vert \dot{\gamma}_{j}^{m}(t) \bigr\vert ^{k-2} \dot{\gamma}_{j}^{m}(t) \int _{\Omega} \vert w_{i} \vert ^{2} \,dx \\ & = \bigl\vert \dot{\gamma}_{j}^{m}(t) \bigr\vert ^{k-2} \dot{\gamma}_{j}^{m}(t) . \end{aligned}
(18)

Then, we insert (16)–(18) in (15) so that (15) yields the following Cauchy problem for a linear ordinary differential equation for unknown functions $$\gamma _{i}^{m}(t)$$ for $$i=1,2, \ldots, m$$;

$$\begin{gathered} \ddot{\gamma}_{i}^{m}(t)+ \gamma _{i}^{m}(t) \lambda _{i}+\nabla \dot{\gamma}_{j}^{m}(t)+ \bigl\vert \dot{ \gamma}_{j}^{m}(t) \bigr\vert ^{k-2} \dot{ \gamma}_{j}^{m}(t)=G_{i}(t), \\ \gamma _{i}^{m}(0)= \int _{\Omega} u_{0} w_{i} \,dx , \qquad \dot{\gamma}_{i}^{m}(0)= \int _{\Omega} u_{1} w_{i} \,dx , \end{gathered}$$
(19)

where

$$G_{i}(t)= \int _{\Gamma _{1}}|u|^{p-2}u\ln |u|w_{i},\quad i=1,2, \ldots, m,$$
(20)

for $$t \in [0, T]$$. Then the problem above has a unique local solution $$\gamma _{i}^{m} \in C^{2}[0, T]$$ for all i, which satisfies a unique $$v_{m}$$ defined by (14) and satisfies (15).

Now, taking $$w_{i}=v_{m t}$$ in equation (15) and then integrating over $$[0, t], 0< t< t_{m}$$ and by parts,

$$\begin{gathered} \bigl\Vert v_{m t}(t) \bigr\Vert ^{2}+ \bigl\Vert \nabla v_{m}(t) \bigr\Vert ^{2}+2 \int _{0}^{t} \bigl\Vert v_{m t}(\tau ) \bigr\Vert _{k, \Gamma _{1}}^{k} \,d\tau \\ \quad = \bigl\Vert v_{1 m}(t) \bigr\Vert ^{2}+ \Vert \nabla v_{0 m} \Vert ^{2}+2 \int _{0}^{t} \int _{\Gamma _{1}}|u|^{p-2} u \ln |u| v_{m t} \,dx \,ds \end{gathered}$$
(21)

for each $$m \geq 1$$.

To estimate the last term on the right-hand side of (21), set $$v_{m} \in H^{1} (0, t_{m} ; H_{\Gamma _{0}}^{1}(\Omega ) )$$ and by the trace theorem; $$v_{m} \in H^{1} (0, t_{m} ; L^{k} (\Gamma _{1} ) )$$. Applying the Young and the trace Sobolev inequalities, we conclude that

\begin{aligned} & 2 \int _{0}^{t} \int _{\Gamma _{1}}|u|^{p-2} u \ln |u| v_{m t} \,dx \,ds \\ &\quad \leq 2 \int _{0}^{t} \int _{\Gamma _{1}} \bigl\vert \vert u \vert ^{p-1} \ln \vert u \vert \bigr\vert \bigl\vert v_{m t}(s) \bigr\vert \,dx \,ds \\ &\quad \leq \int _{0}^{t} \int _{\Gamma _{1}} \vert \vert u \vert ^{p-1} \ln \vert u \vert ^{\frac{k}{k-1}} \,dx \,ds + \int _{0}^{t} \bigl\Vert v_{m t}(s) \bigr\Vert _{k, \Gamma _{1}} \,ds , \end{aligned}
(22)

since $$\Gamma _{1}$$ is bounded. To estimate (22), we focus on the first term

$$\int _{0}^{t} \int _{\Gamma _{1}}\bigl|| u |^{p-1} \ln |u| \bigr|^{\frac{k}{k-1}} \,dx \,ds .$$

We define

$$\Gamma _{1}^{-}=\bigl\{ x\in \Omega ;|u(x)|< 1\bigr\} \quad \text{and}\quad \Gamma _{1}^{+}= \bigl\{ x\in \Omega ;|u(x)|\geq 1\bigr\} ,$$

where $$\Gamma _{1}=\Gamma _{1}^{-} \cup \Gamma _{1}^{+}$$. Because of that, $$\int _{0}^{t} \int _{\Omega}|| u |^{p-1} \ln |u| |^{\frac{k}{k-1}} \,dx \,ds$$ can be recalled as follows

\begin{aligned} &\int _{\Gamma _{1}} \bigl\vert \bigl\vert u(s) \bigr\vert ^{p-1}\ln \bigl|u(s)\bigr| \bigr\vert ^{\frac{k}{k-1}}\,dx \\ &\quad = \int _{\Gamma _{1}^{-}} \bigl\vert \bigl\vert u(s) \bigr\vert ^{p-1}\ln \bigl\vert u(s) \bigr\vert \bigr\vert ^{\frac{k}{k-1}}\,dx + \int _{\Gamma _{1}^{+}} \bigl\vert \bigl\vert u(s) \bigr\vert ^{p-1} \ln \bigl\vert u(s) \bigr\vert \bigr\vert ^{\frac{k}{k-1}}\,dx . \end{aligned}
(23)

Then, the use of Lemma 4 gives

$$\Gamma _{1}^{-} \bigl\vert \bigl\vert u(s) \bigr\vert ^{p-1}\ln \bigl\vert u(s) \bigr\vert \bigr\vert ^{\frac{k}{k-1}}\,dx \leq {}\bigl[ e(p-1)\bigr]^{-\frac{k}{k-1}} \vert \Gamma _{1} \vert =C,$$
(24)

where

$$\inf_{s\in (0,1)}s^{p-1}\ln s=\bigl[e(p-1) \bigr]^{-1}.$$

Let

$$\theta =\frac{2(n-1)}{n-2}\cdot \frac{k}{k-1}-p+1>0\quad \text{for }n \geq 3; \text{each positive }\theta \text{ for }n=1,2.$$

By the Sobolev embedding $$H_{0}^{1}(\Omega )\hookrightarrow L^{\frac{2(n-1)}{n-2}} ( \Gamma _{1} )$$, recalling $$u\in \digamma =C ( [0,T];H_{0}^{1}(\Omega ) )$$, we obtain

\begin{aligned} \int _{\Gamma _{1}^{+}} \bigl\vert \bigl\vert u ( s ) \bigr\vert ^{p-1}\ln \bigl\vert u ( s ) \bigr\vert \bigr\vert ^{\frac{k}{k-1}}\,dx & \leq \int _{\Gamma _{1}^{+}}\theta ^{- \frac{k}{k-1}}\theta ^{\frac{k}{k-1}} \bigl( \bigl\vert u ( s ) \bigr\vert ^{p-1}\ln \bigl\vert u ( s ) \bigr\vert \bigr) ^{\frac{k}{k-1}}\,dx \\ & \leq \theta ^{-\frac{k}{k-1}} \int{\Gamma _{1}^{+}} \bigl( \bigl\vert u ( s ) \bigr\vert ^{p-1}\ln \bigl\vert u ( s ) \bigr\vert ^{\theta } \bigr) ^{\frac{k}{k-1}}\,dx \\ & \leq \theta ^{-\frac{k}{k-1}} \int _{\Gamma _{1}^{+}} \bigl( \bigl\vert u ( s ) \bigr\vert ^{p-1+\theta } \bigr) ^{ \frac{k}{k-1}}\,dx \\ & \leq \theta ^{-\frac{k}{k-1}} \int _{\Gamma _{1}^{+}} \bigl\vert u ( s ) \bigr\vert ^{\frac{2 ( n-1 ) }{n-2}}\,dx \\ & \leq \theta ^{-\frac{k}{k-1}} \int _{\Gamma _{1}=\Gamma _{1}^{-} \cup \Gamma _{1}^{+}} \bigl\vert u ( s ) \bigr\vert ^{ \frac{2 ( n-1 ) }{n-2}}\,dx \\ & = \theta ^{-\frac{k}{k-1}} \Vert u \Vert _{ \frac{2 ( n-1 ) }{n-2}}^{\frac{2 ( n-1 ) }{n-2}} \\ & \leq C \Vert u \Vert _{\digamma }^{ \frac{2 ( n-1 ) }{n-2}}\leq C. \end{aligned}
(25)

Case $$n=1,2$$ proof is similar. So that, for taking $$t=T$$, we conclude that $$|u(s)|^{p-1} \ln |u(s)|$$ is bounded in $$L^{\frac{k}{k-1}} ((0, T) \times \Gamma _{1} )$$.

Writing (24), (25) into (22), we conclude that

$$2 \int _{0}^{t} \int _{\Gamma _{1}}|u(s)|^{p-2}u(s)\ln |u(s)|v_{mt}(s)\,dx\,ds \leq CT+ \int _{0}^{t} \bigl\Vert v_{mt}(s) \bigr\Vert _{k,\Gamma _{1}}\,ds .$$
(26)

Replacing (26) into (21), we can write

$$\begin{gathered} \bigl\Vert v_{m t}(t) \bigr\Vert ^{2}+ \bigl\Vert \nabla v_{m}(t) \bigr\Vert ^{2}+2 \int _{0}^{t} \bigl\Vert v_{m t}(\tau ) \bigr\Vert _{k, \Gamma _{1}}^{k} \,d\tau \\ \quad \leq \bigl\Vert v_{1 m}(t) \bigr\Vert ^{2}+ \Vert \nabla v_{0 m} \Vert ^{2}+C T+ \int _{0}^{t} \bigl\Vert v_{m t}(s) \bigr\Vert _{k, \Gamma _{1}} \,ds , \end{gathered}$$
(27)

where C is a positive constant independent of m. Since the elementary estimate

$$x^{a}\leq C_{1}+C_{2}x\quad \Rightarrow\quad x\leq ( 1+C_{1}+C_{2} ) ^{ \frac{1}{a-1}}$$
(28)

for $$C_{1}, C_{2} \geq 0$$ and $$a>1$$, (27) can be written as

\begin{aligned} & \bigl\Vert v_{m t}(t) \bigr\Vert ^{2}+ \bigl\Vert \nabla v_{m}(t) \bigr\Vert ^{2}+2 \int _{0}^{t} \bigl\Vert v_{m t}(\tau ) \bigr\Vert _{k, \Gamma _{1}}^{k} \,d\tau \\ &\quad \leq C_{4} \end{aligned}
(29)

where $$C_{4}= \Vert v_{1 m}(t) \Vert ^{2}+ \Vert \nabla v_{0 m} \Vert ^{2}+C T+ (1+C_{1}+C_{2} )^{\frac{1}{k-1}}$$. Since

$$\bigl\Vert v_{m}(t) \bigr\Vert \leq \bigl\Vert v_{m}(0) \bigr\Vert +T \Vert v_{mt} \Vert _{L^{\infty } ( 0,T;L^{2}( \Omega ) ) }$$
(30)

we have that $$v_{m}(t)$$ is bounded in $$L^{\infty} (0, T ; H_{\Gamma _{0}}^{1}(\Omega ) )$$. Consequently, it follows from (29) and (30) that

$$\textstyle\begin{cases} v_{m},&\text{is bounded in }\quad L^{\infty } ( 0,T;H_{\Gamma _{0}}^{1}( \Omega ) ) , \\ v_{mt},&\text{is bounded in }L^{\infty } ( 0,T;L^{2}(\Omega ) ) \cap L^{k} ( 0,T;L^{k} ( \Gamma _{1} ) ) .\end{cases}$$
(31)

Using a standard procedure of the Aubin-Lions lemma [30, 34], we deduce that

$$\textstyle\begin{cases} v_{m}\overset{z^{\ast }}{\rightarrow }v & \text{in }L^{\infty } ( 0,T;H_{\Gamma _{0}}^{1}(\Omega ) ) , \\ v_{mt}\overset{z^{\ast }}{\rightarrow }\eta _{1} & \text{in }L^{ \infty } ( 0,T;L^{2}(\Omega ) ) , \\ v_{mt}\overset{z}{\rightarrow }\eta _{2} & \text{in }L^{k} ( (0,T) \times \Gamma _{1} ) , \\ \vert v_{mt} \vert ^{k-2}v_{mt}\overset{z}{\rightarrow } \vert v_{mt} \vert ^{k-2}v_{mt} & \text{in }L^{ \frac{k}{k-1}} ( (0,T)\times \Gamma _{1} ) ,\end{cases}$$

where $$\eta _{1}=v_{t}$$ and $$v(0)=v_{0}$$. Now, we suppose that $$\eta _{2}=v_{t}$$ a.e. in $$(0, T) \times \Gamma _{1}$$. It is clear that, since the weak limit of $$v_{m t}$$ on $$(0, T) \times \partial \Omega$$ is equal to $$\eta _{2}$$ on $$(0, T) \times \Gamma _{1}$$ and to 0 on $$(0, T) \times \Gamma _{0}$$, and since $$u=0$$ on $$(0, T) \times \Gamma _{0}$$, the assumption is that the weal limit of $$v_{m t}$$ on $$(0, T) \times \partial \Omega$$ is the distribution time derivative of v on $$(0, T) \times \partial \Omega$$. Therefore, up to subsequence, we can pass to limit in (15) and find a weak solution (9) applying argument similar to that given in [35] (see Proposition 1).

Uniqueness proof is given by contradiction, claiming two distinct solutions exist. Say w and v have the same initial data. Subtracting both two equations and testing result by $$w_{t}-v_{t}$$, we conclude that

\begin{aligned} & \Vert w_{t}-v_{t} \Vert ^{2}+\|\nabla w-\nabla v\|^{2} \\ &\quad \quad{} + 2 \int _{0}^{T} \int _{\Gamma _{1}} \bigl( \vert w_{\tau} \vert ^{k-2} w_{\tau}- \vert v_{\tau} \vert ^{k-2} v_{\tau} \bigr) (w_{\tau}-v_{ \tau} ) \,d\tau \\ &\quad = 0 . \end{aligned}
(32)

From the following inequality

$$\bigl( |f|^{k-2}f-|g|^{k-2}g \bigr) (f-g)\geq C|f-g|^{k} \quad \text{for }k\geq 2,\forall f,g\in R,\exists C>0$$

equation (9) yields

\begin{aligned} & \Vert w_{t}-v_{t} \Vert ^{2}+\|\nabla w-\nabla v\|^{2} \\ &\quad \quad{} + c \biggl(+ \int _{0}^{T} \Vert \nabla w_{\tau}-\nabla v_{\tau} \Vert _{k, \Gamma _{1}}^{k} \biggr) \\ &\quad \leq 0 \end{aligned}

which satisfies $$w-v=0$$. Therefore, (9) satisfies a unique weak solution. □

Now, we can deal with the proof of Theorem 5.

Proof

To obtain the proof, we apply the contraction mapping theorem. For $$T>0$$, we denote the convex closed subset of $$Y_{T}$$ as

$$X_{T}= \bigl\{ ( v,v_{t} ) \in Y_{T}:v(0,x)=u_{0}(x),v_{t}(0,x)=u_{1}(x) \bigr\} .$$

We define

$$B_{r} ( X_{T} ) = \bigl\{ v\in X_{T}:\bigl\Vert (v)\bigr\Vert _{X_{T}}^{2} \leq r^{2} \bigr\} ,$$

where $$r^{2}=\frac{1}{2} ( \Vert u_{1} \Vert ^{2}+ \Vert \nabla u_{0} \Vert ^{2} )$$. Thanks to Lemma 6, for any $$u \in B_{r} (X_{T} )$$, we can introduce $$v=\zeta (u)$$, which is the unique solution of (9). We can see that v corresponds to u and $$\zeta : X_{T} \rightarrow X_{T}$$. Our aim is to get that ζ is a contraction map, which implies $$\zeta (B_{r} (X_{T} ) ) \subset B_{r} (X_{T} )$$ for any $$T>0$$. Using energy identity for all $$t \in (0, T]$$, we have

\begin{aligned} & \frac{1}{2} \bigl( \Vert v_{t} \Vert ^{2}+\|\nabla v\|^{2} \bigr)+ \int _{0}^{t} \bigl\Vert v_{t}( \tau ) \bigr\Vert _{k, \Gamma _{1}}^{k} \,d\tau \\ &\quad \leq \frac{1}{2} \bigl( \Vert u_{1} \Vert ^{2}+ \Vert \nabla u_{0} \Vert ^{2} \bigr) \\ &\quad \quad{} + \int _{0}^{t} \int _{\Gamma _{1}}|u(s)|^{p-2} u(s) \ln |u(s)| v_{t}(s) \,dx \,ds . \end{aligned}
(33)

Then by

$$\int _{\Gamma _{1}}|u|^{p-2}u\ln |u|\leq \int _{\Gamma _{1}}|u|^{p}$$

(33) yields that

\begin{aligned} & \frac{1}{2} \bigl( \Vert v_{t} \Vert ^{2}+\|\nabla v\|^{2} \bigr)+ \int _{0}^{t} \bigl\Vert v_{t}( \tau ) \bigr\Vert _{k, \Gamma _{1}}^{k} \,d\tau \\ &\quad \leq \frac{1}{2} \bigl( \Vert u_{1} \Vert ^{2}+ \Vert \nabla u_{0} \Vert ^{2} \bigr)+ \int _{0}^{t} \int _{\Gamma _{1}}|u|^{p} v_{t} \,dx \,ds . \end{aligned}
(34)

The last term on the right-hand side of inequality (34) can be estimated using the Holder inequality and similar calculations as for (23) and (25),

\begin{aligned} & \frac{1}{2} \bigl( \Vert v_{t} \Vert ^{2}+\|\nabla v\|^{2} \bigr)+ \int _{0}^{t} \bigl\Vert v_{t}( \tau ) \bigr\Vert _{k, \Gamma _{1}}^{k} \,d\tau \\ &\quad \leq \frac{1}{2} \bigl( \Vert u_{1} \Vert ^{2}+ \Vert \nabla u_{0} \Vert ^{2} \bigr)+C r^{\frac{2(n-1)}{n-2}} T^{\frac{k-1}{k}} \Vert v_{t} \Vert _{L^{k} ((0, T) \times \Gamma _{1} )} . \end{aligned}
(35)

By taking $$t=T$$ and using the inequality (28), we have

$$\Vert v_{t} \Vert _{L^{k} ( (0,T)\times \Gamma _{1} ) }\leq C_{1} \biggl( 1+\frac{1}{2}r_{0}^{2}+Cr^{ \frac{2(n-1)}{n-2}}T^{\frac{k-1}{k}} \biggr) ^{\frac{1}{k-1}}.$$
(36)

Because of the inequality for $$X, Y \geq 0$$,

$$(X+Y)^{a}\leq 2^{a-1} \bigl( X^{a}+Y^{a} \bigr),$$
(37)

where a is a positive constant, (36) yields that

$$\Vert v_{t} \Vert _{L^{k} ( (0,T)\times \Gamma _{1} ) }\leq C_{1} \biggl( 1+\frac{1}{2}r_{0}^{\frac{2}{k-1}}+{Cr}^{ \frac{2(n-1)}{(n-2)(k-1)}}T^{\frac{1}{k}} \biggr) .$$
(38)

Now, we insert (38) into (35) and obtain the following inequality

\begin{aligned} & \frac{1}{2} \bigl( \Vert v_{t} \Vert ^{2}+\|\nabla v\|^{2} \bigr) \\ &\quad \leq \frac{1}{2} r_{0}^{2}+C_{5} r^{\frac{2(n-1)}{n-2}} T^{ \frac{k-1}{k}} \biggl(1+\frac{1}{2} r_{0}^{\frac{2}{k-1}}+C^{ \frac{2(n-1)}{(n-2)(k-1)}} T^{\frac{1}{k}} \biggr) . \end{aligned}
(39)

So that, we have

$$\Vert v_{t} \Vert _{L^{\infty } ( 0,T;L^{2}(\Omega ) ) }\leq r_{0}^{2}+C_{6}r^{\frac{2(n-1)}{n-2}}T^{\frac{k-1}{k}} \biggl( 1+\frac{1}{2}r_{0}^{\frac{2}{k-1}}+Cr^{\frac{2(n-1)}{(n-2)(k-1)}}T^{ \frac{1}{k}} \biggr) .$$
(40)

Using inequality (37) and (40), we have

\begin{aligned} \|v\|_{2}^{2} & \leq \bigl( \Vert u_{0} \Vert _{2}+{ }_{0}^{t} \Vert v_{t} \Vert _{2} \bigr)^{2} \\ & \leq 2 \Vert u_{0} \Vert _{2}^{2}+2 T^{2} \Vert v_{t} \Vert _{L^{ \infty} (0, T ; L^{2}(\Omega ) )} \\ & \leq 2 r_{0}^{2}+2 T^{2} \Vert v_{t} \Vert _{L^{\infty} (0, T ; L^{2}(\Omega ) )} \\ & \leq 2 r_{0}^{2}+2 T^{2} \biggl[r_{0}^{2}+C_{6} r^{ \frac{2(n-1)}{n-2}} T^{\frac{k-1}{k}} \biggl(1+\frac{1}{2} r_{0}^{ \frac{2}{k-1}}+C r^{\frac{2(n-1)}{(n-2)(k-1)}} T^{\frac{1}{k}} \biggr) \biggr] \\ & \leq 2 \bigl(1+T^{2} \bigr) r_{0}^{2}+2 C_{6} T^{\frac{3 k-1}{k}} r^{ \frac{2(n-1)}{n-2}} \biggl(1+ \frac{1}{2} r_{0}^{\frac{2}{k-1}}+ \operatorname{Cr}^{\frac{2(n-1)}{(n-2)(k-1)}} T^{\frac{1}{k}} \biggr) . \end{aligned}
(41)

Combining (39) and (41), we have

$$\Vert v_{t} \Vert _{L^{\infty } ( 0,T;H_{\Gamma _{0}}^{1}( \Omega ) ) }\leq \bigl( 3+T^{2} \bigr) r_{0}^{2}+C_{7}T^{\frac{k-1}{k}}r^{\frac{2(n-1)}{n-2}} \bigl( 1+T^{2} \bigr) \biggl( 1+ \frac{1}{2}r_{0}^{\frac{2}{k-1}}+{Cr}^{\frac{2(n-1)}{(n-2)(k-1)}}T^{ \frac{1}{k}} \biggr) .$$

By choosing T small enough and r large enough, we derive that $$\zeta (u) \in B_{r} (X_{T} )$$ and T= $$T (r_{0}^{2}, k, p, \Omega , \Gamma _{1} )$$ is a decreasing with respect to the first variable.

Next, we will verify that ζ is a contraction mapping continuous on $$B_{r} (X_{T} )$$ and ζ is compact in $$Y_{T}$$. Let $$u_{1}, u_{2} \in X_{r_{0}, T}$$. We define $$v_{1}=\zeta (u_{1} )$$, $$v_{2}=\zeta (u_{2} )$$ with $$u_{1}, u_{2} \in B_{r} (X_{T} )$$, and $$z=v_{1}-v_{2}$$, then, clearly z is a solution of the problem

$$\textstyle\begin{cases} z_{tt}-\Delta z=0 & \text{in }(0,T)\times \Omega, \\ z=0 & \text{on }[0,T)\times \Gamma _{0}, \\ \frac{\partial }{\partial n}z(x,t)=- \vert v_{1t} \vert ^{k-2}v_{1t}+ \vert v_{2t} \vert ^{k-2}v_{2t} & \text{on }[0,T)\times \Gamma _{1}, \\ \hphantom{\frac{\partial }{\partial n}z(x,t)={}}{}+ \vert u_{1} \vert ^{p-2}u_{1}\ln \vert u_{1} \vert - \vert u_{2} \vert ^{p-2}u_{2}\ln \vert u_{2} \vert & \text{on }\Omega , \\ z(0,x)=z_{t}(0,x)=0 . \end{cases}$$
(42)

Since $$v_{1 t}, v_{2 t} \in L^{m} ((0, T) \times \Gamma _{1} )$$, it is clearly that $$\vert v_{1 t} \vert ^{k-2} v_{1 t}$$ and $$\vert v_{2 t} \vert ^{k-2} v_{2 t}$$ belong to $$L^{\frac{k}{k-1}} ((0, T) \times \Gamma _{1} )$$. Also, the functions $$\vert u_{1} \vert ^{p-2} u_{1} \ln \vert u_{1} \vert$$ and $$\vert u_{2} \vert ^{p-2} u_{2} \ln \vert u_{2} \vert$$ belong to $$L^{\frac{k}{k-1}} ((0, T) \times \Gamma _{1} )$$. Then, by using Lemma 6, the energy functional can be written for problem (42) such that

\begin{aligned} & \frac{1}{2} \Vert z_{t} \Vert ^{2}+\frac{1}{2}\|\nabla z\|^{2}+{ }_{0}^{t} \int _{\Gamma _{1}} \bigl( \vert v_{1 t} \vert ^{k-2} v_{1 t}- \vert v_{2 t} \vert ^{k-2} v_{2 t} \bigr) (v_{1 t}-v_{2 t} ) \\ & \quad = \int _{0}^{t} \int _{\Gamma _{1}} \bigl( \vert u_{1} \vert ^{p-2} u_{1} \ln \vert u_{1} \vert - \vert u_{2} \vert ^{p-2} u_{2} \ln \vert u_{2} \vert \bigr) (v_{1 t}-v_{2 t} ) \,dx \,ds \end{aligned}
(43)

for $$0 \leq t \leq T$$. We denote the basic inequality for $$x \geq 2, a_{1}$$, $$a_{2} \in R$$ such that

$$\bigl( \vert a_{1} \vert ^{b-2}a_{1}- \vert a_{2} \vert ^{b-2}a_{2} \bigr) ( a_{1}-a_{2} ) \geq C^{ \ast } \vert a_{1}-a_{2} \vert ^{b}.$$

For estimating the last integral on the left-hand side of (43), we apply the basic inequality by taking $$b=k$$ when $$k \geq 2$$ and $$b=\frac{k}{k-1}$$ when $$1< k<2$$. So that, (43) becomes

\begin{aligned} & \frac{1}{2} \Vert z_{t} \Vert ^{2}+\frac{1}{2}\|\nabla z\|^{2}+C^{*} \Vert z_{t} \Vert _{L^{k} ((0, T) \times \Gamma _{1} )}^{k} \\ &\quad = \int _{0}^{t} \int _{\Gamma _{1}} \bigl( \vert u_{1} \vert ^{p-2} u_{1} \ln \vert u_{1} \vert - \vert u_{2} \vert ^{p-2} u_{2} \ln \vert u_{2} \vert \bigr) (v_{1 t}-v_{2 t} ) \,dx \,ds \end{aligned}
(44)

for $$k \geq 2$$, and

\begin{aligned} & \frac{1}{2} \Vert z_{t} \Vert ^{2}+\frac{1}{2}\|\nabla z\|^{2}+C^{*} \bigl\Vert \vert v_{1 t} \vert ^{k-2} v_{1 t}- \vert v_{2 t} \vert ^{k-2} v_{2 t} \bigr\Vert _{L^{\frac{k}{k-1}} ((0, T) \times \Gamma _{1} )}^{\frac{k}{k-1}} \\ &\quad = \int _{0}^{t} \int _{\Gamma _{1}} \bigl( \vert u_{1} \vert ^{p-2} u_{1} \ln \vert u_{1} \vert - \vert u_{2} \vert ^{p-2} u_{2} \ln \vert u_{2} \vert \bigr) (v_{1 t}-v_{2 t} ) \,dx \,ds \end{aligned}
(45)

for $$1< k<2$$.

Now, we need to estimate the logarithmic term in (45). If we set

$$G(s)=|s|^{p-2}s\ln |s|,$$

then

\begin{aligned} G^{\prime}(s) & =(p-1)|s|^{p-2} \ln |s|+|s|^{p-2} \\ & =\bigl(1+(p-1) \ln |s|\bigr)|s|^{p-2} . \end{aligned}

From the mean value theorem, we have

\begin{aligned} &\bigl\vert G (u_{1} )-G (u_{2} ) \bigr\vert \\ &\quad = \bigl\vert G^{ \prime} \bigl( \vartheta u_{1}+(1-\vartheta ) u_{2} \bigr) (u_{1}-u_{2} ) \bigr\vert \\ &\quad \leq \bigl[1+(p-1) \ln \bigl\vert \bigl(\vartheta u_{1}+(1- \vartheta ) u_{2} \bigr) \bigr\vert \bigr] \bigl\vert \bigl( \vartheta u_{1}+(1- \vartheta ) u_{2} \bigr) \bigr\vert ^{p-2} \vert u_{1}-u_{2} \vert , \end{aligned}

where $$0<\vartheta <1$$. From Lemma 4, we conclude that

\begin{aligned} \bigl\vert G (u_{1} )-G (u_{2} ) \bigr\vert & \leq \bigl\vert \bigl(\vartheta u_{1}+(1-\vartheta ) u_{2} \bigr) \bigr\vert ^{p-2} \vert u_{1}-u_{2} \vert +(p-1) A \vert u_{1}-u_{2} \vert \\ &\quad{} + (p-1) \vert u_{1}-u_{2} \vert \bigl\vert \bigl( \vartheta u_{1}+(1- \vartheta ) u_{2} \bigr) \bigr\vert ^{p-2+\varepsilon} \\ & \leq \bigl\vert (u_{1}+u_{2} ) \bigr\vert ^{p-2} \vert u_{1}-u_{2} \vert +(p-1) A \vert u_{1}-u_{2} \vert \\ &\quad{} + (p-1) \vert u_{1}-u_{2} \vert \vert u_{1}+u_{2} \vert ^{p-2+ \varepsilon} .\end{aligned}
(46)

Inserting (47) into (45), we obtain

\begin{aligned} & \frac{1}{2} \Vert z_{t} \Vert ^{2}+\frac{1}{2}\|\nabla z\|^{2}+C^{*} \bigl\Vert \vert v_{1 t} \vert ^{k-2} v_{1 t}- \vert v_{2 t} \vert ^{k-2} v_{2 t} \bigr\Vert _{L^{\frac{k}{k-1}}}^{\frac{k}{k-1}} \bigl((0, T) \times \Gamma _{1} \bigr) \\ &\quad = \int _{0}^{T} \int _{\Gamma _{1}}\begin{pmatrix} \vert (u_{1}+u_{2} ) \vert ^{p-2} \vert u_{1}-u_{2} \vert +(p-1) A \vert u_{1}-u_{2} \vert \\ \quad +(p-1) \vert u_{1}-u_{2} \vert \vert u_{1}+u_{2} \vert ^{p-2+ \varepsilon} \end{pmatrix} (v_{1 t}-v_{2 t} ) \,dx \,ds. \end{aligned}
(47)

We choose $$\varkappa _{0} \in (p, \varkappa )$$ such that

$$\frac{\varkappa }{\varkappa -p+1}< \frac{\varkappa _{0}}{\varkappa _{0}-p+1}< k.$$
(48)

Using (49), we can define $$l \in (0,1)$$ such that

$$\frac{1}{k}+\frac{1}{\varkappa }+\frac{1}{l}=1,$$
(49)

where $$l<\frac{\varkappa _{0}}{p-2}$$.

Using (37) and the Holder inequality, we can write the first term of the integral term of (48) as

\begin{aligned} & \int _{0}^{T} \int _{\Gamma _{1}} \bigl\vert (u_{1}+u_{2} ) \bigr\vert ^{p-2} \vert u_{1}-u_{2} \vert (v_{1 t}-v_{2 t} ) \\ &\quad \leq 2^{l-1} \int _{0}^{T} \Vert u_{1}-u_{2} \Vert _{\varkappa _{0}, \Gamma _{1}} \bigl( \Vert u_{1} \Vert _{l(p-2), \Gamma _{1}}^{(p-2)}+ \Vert u_{2} \Vert _{l(p-2), \Gamma _{1}}^{(p-2)} \bigr) \Vert z_{t} \Vert _{k, \Gamma _{1}} . \end{aligned}
(50)

Since $$l(p-2)<\varkappa _{0}$$, by the trace Sobolev embedding and definition of r, we obtain

\begin{aligned} & \int _{0}^{T} \int _{\Gamma _{1}} \bigl\vert (u_{1}+u_{2} ) \bigr\vert ^{p-2} \vert u_{1}-u_{2} \vert (v_{1 t}-v_{2 t} ) \\ &\quad \leq 2^{l-1} \int _{0}^{T} \Vert u_{1}-u_{2} \Vert _{\varkappa _{0}, \Gamma _{1}} \bigl( \Vert \nabla u_{1} \Vert _{2}^{(p-2)}+ \Vert \nabla u_{2} \Vert _{2}^{(p-2)} \bigr) \Vert z_{t} \Vert _{k, \Gamma _{1}} \\ &\quad \leq 2 C_{10} r^{p-2} \int _{0}^{T} \Vert u_{1}-u_{2} \Vert _{ \varkappa _{0}, \Gamma _{1}} \Vert z_{t} \Vert _{k, \Gamma _{1}} . \end{aligned}
(51)

Applying the Holder inequality, we conclude that

\begin{aligned} & \int _{0}^{T} \int _{\Gamma _{1}} \bigl\vert (u_{1}+u_{2} ) \bigr\vert ^{p-2} \vert u_{1}-u_{2} \vert (v_{1 t}-v_{2 t} ) \\ &\quad \leq 2 C_{10} r^{p-2} T^{\frac{k-1}{k}} \Vert u_{1}-u_{2} \Vert _{L^{ \infty} (0, T ; L^{\varkappa} (\Gamma _{1} ) )} \Vert z_{t} \Vert _{L^{k} ((0, T) \times (\Gamma _{1} ) )} . \end{aligned}
(52)

Thanks to (40) and $$r_{0} \leq r$$, (52) yields

\begin{aligned} & \int _{0}^{T} \int _{\Gamma _{1}} \bigl\vert (u_{1}+u_{2} ) \bigr\vert ^{p-2} \vert u_{1}-u_{2} \vert (v_{1 t}-v_{2 t} ) \\ &\quad \leq C_{11} r^{p-2} \biggl[ \biggl(1+ \frac{1}{2} r^{\frac{2}{k-1}} \biggr) T^{\frac{k-1}{k}}+ \operatorname{Cr}^{ \frac{2(n-1)}{(n-2)(k-1)}} T \biggr] \\ &\quad \quad{} \times \Vert u_{1}-u_{2} \Vert _{L^{\infty} (0, T ; L^{ \varkappa _{0}} (\Gamma _{1} ) )} . \end{aligned}
(53)

If we choose $$\varkappa _{1} \in (p, \varkappa )$$ such that

$$\frac{\varkappa }{\varkappa -(p+\varepsilon )+1}< \frac{\varkappa _{1}}{\varkappa _{1}-(p+\varepsilon )+1}< k.$$
(54)

Using (54), we can define $$l_{1} \in (0,1)$$ such that

$$\frac{1}{k}+\frac{1}{\varkappa }+\frac{1}{l_{1}}=1,$$

where $$l_{1}<\frac{\varkappa _{1}}{p-2+\varepsilon}$$. Using calculations similar to (50)–(52), we obtain

\begin{aligned} & \int _{0}^{T} \int _{\Gamma _{1}} \vert u_{1}-u_{2} \vert \vert u_{1}+u_{2} \vert ^{p-2+\varepsilon} (v_{1 t}-v_{2 t} ) \\ &\quad \leq C_{12} r^{p+\varepsilon -2} \biggl[ \biggl(1+ \frac{1}{2} r^{ \frac{2}{k-1}} \biggr) T^{\frac{k-1}{k}}+ \operatorname{Cr}^{ \frac{2(n-1)}{(n-2)(k-1)}} T \biggr] \\ &\quad \quad{} \times \Vert u_{1}-u_{2} \Vert _{L^{\infty} (0, T ; L^{ \varkappa} (\Gamma _{1} ) )}, \end{aligned}
(55)

where $$\varepsilon >0$$ constant.

Using the trace Sobolev embedding and the Holder inequality in time and (36), we have

\begin{aligned} &\int _{0}^{T} \int _{\Gamma _{1}} \vert u_{1}-u_{2} \vert (v_{1 t}-v_{2 t} ) \\ &\quad \leq \int _{0}^{T} \Vert u_{1}-w_{u 2} \Vert _{ \varkappa _{3}, \Gamma _{1}} \Vert z_{t} \Vert _{k, \Gamma _{1}} \\ &\quad \leq \Vert u_{1}-u_{2} \Vert _{L^{\infty} (0, T ; L^{ \varkappa} (\Gamma _{1} ) )} \Vert z_{t} \Vert _{L^{k} ((0, T) \times (\Gamma _{1} ) )} \\ &\quad \leq C_{1} \biggl(1+\frac{1}{2} r_{0}^{\frac{2}{k-1}}+ \operatorname{Cr}^{\frac{2(n-1)}{(n-2)(k-1)}} T^{\frac{1}{k}} \biggr) \Vert u_{1}-u_{2} \Vert _{L^{\infty} (0, T ; L^{\varkappa _{3} (\Gamma _{1} )} )}, \end{aligned}
(56)

where $$\varkappa _{3} \in (p, \varkappa )$$.

By combining (56),(55), and (53), we obtain

\begin{aligned} & \int _{0}^{T} \int _{\Gamma _{1}}\begin{pmatrix} \vert (u_{1}+u_{2} ) \vert ^{p-2} (v_{1 t}-v_{2 t} )+(p-1) A (v_{1 t}-v_{2 t} ) \\ +(p-1) (v_{1 t}-v_{2 t} ) \vert u_{1}+u_{2} \vert ^{p-2+ \varepsilon} \end{pmatrix} (v_{1 t}-v_{2 t} ) \,dx \,ds \\ & \quad \leq K \Vert u_{1}-u_{2} \Vert _{L^{\infty} (0, T ; L^{ \varkappa *} (\Gamma _{1} ) )}, \end{aligned}
(57)

where

$$\varkappa ^{\ast }=\max \{ \varkappa _{1},\varkappa _{2}, \varkappa _{3} \}$$

and

$$K=\max \textstyle\begin{cases} C_{1} ( 1+\frac{1}{2}r_{0}^{\frac{2}{k-1}}+Cr^{ \frac{2 ( n-1 ) }{ ( n-2 ) ( k-1 ) }}T^{\frac{1}{k}} ) , \\ C_{12}r^{p+ \varepsilon -2} [ ( 1+\frac{1}{2}r^{\frac{2}{k-1}} ) T^{\frac{k-1}{k}}+Cr^{ \frac{2 ( n-1 ) }{ ( n-2 ) ( k-1 ) }}T ] , \\ C_{11}r^{p-2} [ ( 1+\frac{1}{2}r^{\frac{2}{k-1}} ) T^{ \frac{k-1}{k}}+Cr^{ \frac{2 ( n-1 ) }{ ( n-2 ) ( k-1 ) }}T ] . \end{cases}$$

Consequently, by inserting (57) into (44) and (45), we get the following estimates

\begin{aligned}& \Vert v_{1t}-v_{2t} \Vert _{L^{\infty } ( 0,T;L^{2} ( \Omega ) ) }^{2} \leq K_{1} \Vert u_{1}-u_{2} \Vert _{L^{\infty } ( 0,T;L^{\varkappa ^{\ast }} ( \Gamma _{1} ) ) }, \end{aligned}
(58)
\begin{aligned}& \Vert \nabla v_{1}-\nabla v_{2} \Vert _{L^{\infty } ( 0,T;L^{2} ( \Omega ) ) }^{2}\leq K_{1} \Vert u_{1}-u_{2} \Vert _{L^{\infty } ( 0,T;L^{\varkappa ^{\ast }} ( \Gamma _{1} ) ) }, \end{aligned}
(59)

and

$$\Vert v_{1t}-v_{2t} \Vert _{L^{k} ( (0,T)\times \Gamma _{1} ) }^{k} \leq K_{1} \Vert u_{1}-u_{2} \Vert _{L^{\infty } ( 0,T;L^{\varkappa ^{\ast }} ( \Gamma _{1} ) ) },$$
(60)

where $$k \geq 2$$, while

$$\bigl\Vert \vert v_{1t} \vert ^{k-2}v_{1t}- \vert v_{2t} \vert ^{k-2}v_{2t} \bigr\Vert _{L^{\frac{k}{k-1}} ( (0,T) \times \Gamma _{1} ) }^{\frac{k}{k-1}}\leq K_{1} \Vert u_{1}-u_{2} \Vert _{L^{\infty } ( 0,T;L^{\varkappa ^{\ast }} ( \Gamma _{1} ) ) },$$
(61)

where $$1< k<2$$ and $$K_{1}>0$$ is a constant, which depends on $$(p, k, \Omega , \Gamma _{1}, T, r )$$. Thanks to $$v_{1}(0)=v_{2}(0)=0$$, we conclude that for $$0 \leq t \leq T$$,

$$\Vert v_{1}-v_{2} \Vert _{2}\leq \int _{0}^{T} \Vert v_{1t}-v_{2t} \Vert _{2}\leq T \Vert v_{1t}-v_{2t} \Vert _{L^{\infty } ( 0,T;L^{2}(\Omega ) ) }.$$
(62)

Plug (58) into (62) yields that

$$\Vert v_{1}-v_{2} \Vert _{L^{\infty } ( 0,T;L^{2}( \Omega ) ) }^{2} \leq K_{1}T^{2} \Vert u_{1}-u_{2} \Vert _{L^{\infty } ( 0,T;L^{\varkappa ^{\ast }} ( \Gamma _{1} ) ) }.$$
(63)

Thus, from estimates (58)–(63), we get contractiveness of ζ in $$B_{r} (X_{T} )$$. It follows that $$v=\zeta (u)$$ is a Cauchy sequence in $$Y_{T}$$. The proof is completed. □

4 Potential well

In this section, we will demonstrate the global existence of the proofs of solution (1).

We defined some useful functionals total energy function as

\begin{aligned}& J ( u ) =\frac{1}{2} \Vert \nabla u \Vert ^{2}+ \frac{1}{p^{2}} \Vert u \Vert _{p,\Gamma _{1}}^{p}- \frac{1}{p}\int _{\Gamma _{1}}u^{p}\ln \vert u \vert \,dx , \end{aligned}
(64)
\begin{aligned}& I ( u ) = \Vert \nabla u \Vert ^{2}- \int _{\Gamma _{1}}u^{p}\ln \vert u \vert \,dx . \end{aligned}
(65)

Then, combining (64), (65), and definition of $$E(u)$$ gives

$$J(u)=\frac{1}{p}I(u)+\frac{1}{p^{2}}\Vert u\Vert _{p}^{p}$$
(66)

and

$$E(u)=\frac{1}{2} \Vert u_{t} \Vert ^{2}+J(u).$$
(67)

The potential well depth is defined as

$$W= \bigl\{ ( u_{0},u_{1} ) \in H_{\Gamma _{0}}^{1}( \Omega )\times L^{2}(\Omega ):J(u)\leq d,I(u)>0 \bigr\} \cup \{0\},$$
(68)

and the outer space of the potential well

$$V= \bigl\{ ( u_{0},u_{1} ) \in H_{\Gamma _{0}}^{1}( \Omega )\times L^{2}(\Omega ):J(u)\leq d,I(u)< 0 \bigr\} .$$
(69)

Lemma 8

Let $$u_{0}\in H_{\Gamma _{0}}^{1}(\Omega )\backslash \{0\}$$, $$\Vert u\Vert _{p, \Gamma _{1}}^{p}\neq 0$$. Then

1. i)

$$\lim_{\lambda \rightarrow 0^{+}} J(\lambda u)=0$$, $$\lim_{\lambda \rightarrow \infty} J(\lambda u)=-\infty$$;

2. ii)

There exists $$\lambda ^{*}>0$$ satisfying $$\frac{d}{d \lambda} J (\lambda ^{*} u )=0$$ such that

$$I(\lambda u)=\frac{d}{d\lambda }J(\lambda u)\textstyle\begin{cases} >0, & 0\leq \lambda < \lambda ^{\ast }, \\ =0, & \lambda =\lambda ^{\ast }, \\ < 0, & \lambda < \lambda ^{\ast }< \infty. \end{cases}$$

Proof

i) Take $$J(\lambda u)$$,

\begin{aligned} J(\lambda u) & =\frac{1}{p}\|\lambda \nabla u\|^{2}+\frac{1}{p^{2}}\| \lambda u\|_{p, \Gamma _{1}}^{p}- \frac{1}{p} \int _{\Gamma _{1}} \ln | \lambda u|(\lambda u)^{p} \,dx \\ & =\frac{\lambda ^{2}}{2}\|\nabla u\|^{2}+\frac{\lambda ^{p}}{p^{2}} \|u \|_{p, \Gamma _{1}}^{p}-\frac{\lambda ^{p}}{p} \int _{\Gamma _{1}}|u|^{p} \ln |u| \,dx - \frac{\lambda ^{p}}{p} \ln |\lambda | \int _{\Gamma _{1}}|u|^{p} \,dx . \end{aligned}

By virtue of $$\|u\|_{p, \Gamma _{1}}^{p}$$, we see that $$\lim_{\lambda \rightarrow 0} J(\lambda u)=0$$, $$\lim_{\lambda \rightarrow \infty} J(\lambda u)=-\infty$$.

ii) Now, taking the derivative of $$J(\lambda u)$$ with respect to λ, we have

$$\frac{d}{d\lambda }J(\lambda u)=\lambda \biggl( \Vert \nabla u\Vert ^{2}- \lambda ^{p-2} \int _{\Gamma _{1}}|u|^{p}\ln |u|\,dx -\lambda ^{p-2}\ln | \lambda |\Vert u\Vert _{p,\Gamma _{1}}^{p} \biggr) .$$
(70)

Thanks to definition of $$J(\lambda u)$$, it is clearly from (70) that $$\lambda ^{-1} \frac{d}{d \lambda} J(\lambda u)=N(\lambda u)$$. So, we obtain

\begin{aligned} \frac{d}{d \lambda} N(\lambda u) & =(2-p) \lambda ^{p-3} \int _{ \Gamma _{1}}|u|^{p} \ln |u| \,dx -\lambda ^{p-3}\|u\|_{p, \Gamma _{1}}^{p}+(2-p) \lambda ^{p-3} \ln |\lambda |\|u\|_{p, \Gamma _{1}}^{p} \\ & =\lambda ^{p-3} \biggl((2-p) \int _{\Gamma _{1}}|u|^{p} \ln |u| \,dx - \|u \|_{p, \Gamma _{1}}^{p}+(2-p) \ln |\lambda |\|u\|_{p, \Gamma _{1}}^{p} \biggr) . \end{aligned}

Therefore, there is a unique $$\lambda ^{1}$$ such that $$\frac{d}{d \lambda} N(\lambda u) |_{\lambda =\lambda ^{1}}=0$$, by taking

$$\lambda ^{1}=\exp \biggl( \frac{\Vert u\Vert _{p,\Gamma _{1}}^{p}-(2-p)\int _{\Gamma _{1}}|u|^{p}\ln |u|\,dx }{(2-p)\Vert u\Vert _{p,\Gamma _{1}}^{p}} \biggr) >0$$

such that $$\frac{d}{d \lambda} N(\lambda u)>0$$ on $$(0, \lambda ^{1} )$$ and $$\frac{d}{d \lambda} N(\lambda u)<0$$ on $$(\lambda ^{1}, \infty )$$. Because of $$N(\lambda u) |_{\lambda =0}=$$ $$\|\nabla u\|^{2}>0$$ and $$\lim_{\lambda \rightarrow \infty} N(\lambda u)=-\infty$$, there is one $$\lambda ^{*}>0$$ such that $$N (\lambda ^{*} u )=0$$, i.e $$\frac{d}{d \lambda} J (\lambda ^{*} u )=0$$.

A simple corollary of the fact that

$$\frac{d}{d\lambda }J(\lambda u)=\lambda N(\lambda u)$$

which gives that $$\frac{d}{d \lambda} J(\lambda u)>0$$ on $$(0, \lambda ^{*} )$$ and $$\frac{d}{d \lambda} J(\lambda u)<0$$ on $$(\lambda ^{*}, \infty )$$. Thus, we have the desired results such that

$$I(\lambda u)=\lambda g^{\prime }(\lambda )\textstyle\begin{cases} >0, & 0\leq \lambda < \lambda ^{\ast }, \\ =0, & \lambda =\lambda ^{\ast }, \\ < 0, & \lambda < \lambda ^{\ast }< \infty. \end{cases}$$

□

Lemma 9

i) The depth of potential well depth defined by

$$d=\inf_{u\in H_{\Gamma _{0}}^{1}(\Omega )\backslash \{0\}, u | _{\Gamma _{1}}\neq 0}\sup_{\lambda >0}J(\lambda u).$$
(71)

Then d is a positive function such that

$$0< d=\inf_{u\in N}J(u)$$
(72)

where N is the Nehari manifold given by

$$N= \bigl\{ u\in H_{\Gamma _{0}}^{1}(\Omega )\backslash \{0 \}:I(u)=0 \bigr\} ,$$

and d has a positive lower bound, namely,

$$d\geq \frac{1}{2} \biggl( \frac{e\alpha }{C^{\ast }} \biggr) ^{ \frac{2}{p+\alpha -2}},$$

where C is defined as a positive constant.

Proof

i) By (64), thanks to definitions of the Nehari manifold and d, it satisfies $$d\geq 0$$. So that, our purpose is to prove that there is a positive function such that $$J(u)=d$$. We define $$\{ u_{i} \} _{i=1}^{\infty }\subset N$$ as a minimizing sequence for J. So that, we conclude that

$$\lim_{i\rightarrow \infty }J ( u_{i} ) =d.$$

It is clearly that, $$\{ \vert u_{i} \vert \}_{i=1}^{\infty} \subset N$$ a minimizing sequence for J. Now, we suppose that $$u_{i}>0$$ in Ω for all $$i \in \mathbb{N}$$.

We also obtain that J is coercive on $$u \in N$$ satisfying $$\{u_{i} \}_{i=1}^{\infty}$$ and is bounded in $$H_{\Gamma _{0}}^{1}(\Omega )$$. Since $$H_{\Gamma _{0}}^{1}(\Omega ) \hookrightarrow L^{p+\alpha} ( \Gamma _{1} )$$ is compact embedding, there is a function u and a subsequence of $$\{ \vert u_{i_{n}} \vert \}_{i=1}^{\infty}$$ of $$\{ \vert u_{i} \vert \}_{i=1}^{\infty}$$, such that

\begin{aligned} & u_{i_{n}} \rightarrow u \quad \text{weakly in } H_{\Gamma _{0}}^{1}( \Omega ), \\ & u_{i_{n}} \rightarrow u \quad \text{strongly in } L^{p+\alpha} ( \Gamma _{1} ) , \\ & u_{i_{n}} \rightarrow u \quad \text{a.e. in } \Omega , \end{aligned}

where $$i_{n} \rightarrow \infty$$.

Then, we get $$u \geq 0$$ a.e. in Ω. Moreover, using the dominated convergence theorem, weak lower semicontinuity and definition of $$J(u)$$, $$I(u)$$ and N gives

\begin{aligned} & J(u) \leq \lim_{i_{n} \rightarrow \infty} \inf J (u_{i_{n}} )=d, \\ & I(u) \leq \lim_{i_{n} \rightarrow \infty} \inf I (u_{i_{n}} )=0 . \end{aligned}

Since $$x^{-y} \ln x \leq \frac{1}{e y}$$ for $$x, y>0$$ and the trace Sobolev embbedding theorem, we have

\begin{aligned} \Vert \nabla u_{i_{n}} \Vert ^{2} & = \int _{\Gamma _{1}} \vert u_{i_{n}} \vert ^{p} \ln \vert u_{i_{n}} \vert \\ & \leq \int _{\Gamma _{1}} \frac{1}{e \alpha} \vert u_{i_{n}} \vert ^{p} \ln \vert u_{i_{n}} \vert ^{\alpha} \\ & =\frac{1}{e \alpha} \Vert u_{i_{n}} \Vert _{p+\alpha , \Gamma _{1}}^{p+\alpha} \\ & \leq \frac{C^{*}}{e \alpha} \Vert \nabla u_{i_{n}} \Vert ^{p+ \alpha}, \end{aligned}

where $$C^{*}$$ is the best Sobolev constant, which means

$$\biggl( \frac{e\alpha }{C^{\ast }} \biggr) ^{\frac{2}{p+\alpha -2}}< \Vert \nabla u_{i_{n}} \Vert ^{2}.$$

Therefore, we conclude that

$$\int _{\Gamma _{1}} \vert u_{i_{n}} \vert ^{p}\ln \vert u_{i_{n}} \vert \geq \biggl( \frac{e\alpha }{C^{\ast }} \biggr) ^{\frac{2}{p+\alpha -2}}.$$

Using the dominated convergence theorem, we have

$$\int _{\Gamma _{1}}|u|^{p}\ln |u|\geq \biggl( \frac{e\alpha }{C^{\ast }} \biggr) ^{\frac{2}{p+\alpha -2}\cdot }>0$$

which means that $$u \neq 0$$.

Last, we show that $$I(u)=0$$. Indeed, if it is not true, we get $$I(u)<0$$. So, thanks to Lemma 8, we have a positive constant $$\lambda ^{*}<1$$ implying that $$I (\lambda ^{*} u )=0$$. Therefore, it follows that

\begin{aligned} d & \leq J \bigl(\lambda ^{*} u \bigr)= \frac{ (\lambda ^{*} )^{2}}{2}\|\nabla u\|^{2}+ \frac{ (\lambda ^{*} )^{p}}{p^{2}}\|u \|_{p, \Gamma _{1}}^{p} \\ & = \bigl(\lambda ^{*} \bigr)^{2} \biggl[ \frac{1}{2}\|\nabla u\|^{2}+ \frac{ (\lambda ^{*} )^{p-2}}{p^{2}}\|u \|_{p, \Gamma _{1}}^{p} \biggr] \\ & \leq \bigl(\lambda ^{*} \bigr)^{2} \biggl[ \frac{1}{2}\|\nabla u\|^{2}+ \frac{1}{p^{2}}\|u \|_{p, \Gamma _{1}}^{p} \biggr] \\ & \leq \bigl(\lambda ^{*} \bigr)^{2} \lim _{i_{n} \rightarrow \infty} \inf \biggl[\frac{1}{2} \Vert \nabla u_{i_{n}} \Vert ^{2}+ \frac{1}{p^{2}} \Vert u_{i_{n}} \Vert _{p, \Gamma _{1}}^{p} \biggr] \\ & \leq \bigl(\lambda ^{*} \bigr)^{2} \lim _{k \rightarrow \infty} \inf J (u_{i_{n}} ) \\ & = \bigl(\lambda ^{*} \bigr)^{2} d< d , \end{aligned}

where $$d=\frac{1}{2} (\frac{e \alpha}{C^{*}} )^{ \frac{2}{p+\alpha -2}}$$. it is a contradiction. □

5 Lower bound for blow-up time

In this part, we prove a lower bound for blow-up time of problem (1). First, we give lemma, which will play a role of the proof of Theorem 11.

Lemma 10

Suppose that $$( u_{0},u_{1} ) \in V$$, $$2\leq p<\varkappa$$. So, we get $$( u,u_{t} ) \in V$$ for all $$t\geq 0$$. Proof. By way of contradiction, suppose that $$( u_{0},u_{1} )$$ leaves V at time $$t=t_{0}$$, so there is a sequence $$\{ t_{s} \} ,t_{s}\rightarrow t_{0}^{-}$$ such that $$I ( u ( t_{s} ) ) \leq 0$$ and $$E ( u ( t_{s} ) ) \leq d$$. Thanks to weak lower semicontinuity $$\Vert \cdot \Vert _{H_{\Gamma _{0}}^{1}}$$, we obtain

$$I \bigl( u ( t_{0} ) \bigr) \leq \lim_{n\rightarrow \infty } \inf I \bigl( u ( t_{s} ) \bigr) \leq 0,$$
(73)

and

$$E ( t_{0} ) \leq \lim_{n\rightarrow \infty }\inf E \bigl( u ( t_{s} ) \bigr) \leq d.$$
(74)

If we take $$(u (t_{0} ), u_{t} (t_{0} ) ) \notin V$$, $$I (u (t_{0} ) )=0$$ or $$E (u (t_{0} ) )>d$$. Because of (6), taking $$E (t_{0} )>d$$ is impossible, which is a contradiction with inequality (74). By the continuity of function $$I(u(t))$$ about time, if we take $$I (u (t_{0} ) )=0$$, by definition of d, (64) and (3), we arrive at

$$d\geq E ( t_{0} ) \geq J \bigl( u ( t_{0} ) \bigr) \geq \inf_{u\in N}J(u)=d.$$

Moreover, we have a contradiction. So, we get $$(u, u_{t} ) \in V$$ for all $$t \geq 0$$.

Theorem 11

Assume that $$( u_{0},u_{1} ) \in V$$, $$2\leq p<\varkappa$$ and $$2< p<1+\frac{(2k-2)(n-1)}{k(n-2)}$$. Then, the solutions u of problem (1) are bounded at finite time $$t=T_{1}$$ with

$$\lim_{t\rightarrow T_{1}^{-}} \Vert u_{t} \Vert ^{2}+ \Vert \nabla u\Vert ^{2}=\infty.$$

Therefore, we give lower bound for $$T_{1}$$ such that

$$T_{1}\geq \frac{d\theta }{H(t)+e(p-1)^{-\frac{k}{k-1}} \vert \Gamma _{1} \vert +(e\alpha )^{-\frac{k}{k-1}} ( K_{2} ) ^{\frac{k}{k-1}(p-1+\alpha )}\theta ^{\frac{k}{k-1}(p-1+\alpha )}},$$

where $$0<\alpha <\frac{2 n-2}{n-2}-p, K_{2}$$ is the positive Sobolev constant and $$H(0)= \Vert u_{1} \Vert ^{2}+ \Vert \nabla u_{0} \Vert ^{2}$$.

Proof

We define a function as

$$H(t)= \Vert u_{t} \Vert ^{2}+\Vert \nabla u\Vert ^{2}.$$
(75)

By testing equation of problem (1) by $$u_{t}(x, t)$$ and using the Green formula, we obtain

$$( u_{tt},u_{t} ) =-{ \int}_{\Omega }\nabla u,\nabla u_{t}+{ \int}_{\Gamma _{1}} \bigl[ - \vert u_{t} \vert ^{k-2}u_{t}+|u|^{p-2}u\ln |u| \bigr] u_{t}.$$
(76)

By differentiating (75) and using (76), we conclude that

\begin{aligned} H^{\prime}(t) & =2 (u_{t t}, u_{t} )+2{ \int}_{\Omega} \nabla u \nabla u_{t} \\ & =-2{ \int}_{\Omega} \nabla u, \nabla u_{t}+2{ \int}_{\Gamma _{1}} \bigl[- \vert u_{t} \vert ^{k-2} u_{t}+|u|^{p-2} u \ln |u| \bigr] u_{t}+2{ \int}_{\Omega} \nabla u \nabla u_{t} \\ & =-2{ \int}_{\Gamma _{1}} \vert u_{t} \vert ^{k-1} u_{t}+2{ \int}_{ \Gamma _{1}}|u|^{p-2} u \ln |u| u_{t} . \end{aligned}
(77)

Since $$2< p<1+\frac{(2 k-2)(n-1)}{k(n-2)}$$, by applying the trace Sobolev embedding theorem where α is a positive constant such that $$(p-1+\alpha )<\frac{(2 k-2)(n-1)}{k(n-2)}$$. Therefore, if we use the Young inequality and Sobolev theorems, (77) yields that

\begin{aligned} &2 \int _{\Gamma _{1}} \vert u \vert ^{p-2}u\ln \vert u \vert u_{t} \\ &\quad \leq \Vert u_{t} \Vert _{k,\Gamma _{1}}^{k}+ \biggl( \int _{\Gamma _{1}} \vert u \vert ^{p-2}u\ln \vert u \vert \biggr) ^{\frac{k}{k-1}} \\ &\quad \leq \Vert u_{t} \Vert _{k,\Gamma _{1}}^{k} \int _{ \Gamma _{1}^{-}} \bigl( \vert u \vert ^{p-2}u\ln \vert u \vert \bigr) ^{\frac{k}{k-1}}+ \int _{\Gamma _{1}^{+}} \bigl( \vert u \vert ^{p-2}u\ln \vert u \vert \bigr) ^{\frac{k}{k-1}} \\ &\quad \leq \Vert u_{t} \Vert _{k,\Gamma _{1}}^{k}+e ( p-1 ) ^{-\frac{k}{k-1}} \vert \Gamma _{1} \vert + ( e\alpha ) ^{-\frac{k}{k-1}} \int _{\Gamma _{1}^{+}} \bigl( \vert u \vert ^{p-1+\alpha } \bigr) ^{\frac{k}{k-1}} \\ &\quad \leq \Vert u_{t} \Vert _{k,\Gamma _{1}}^{k}+e ( p-1 ) ^{-\frac{k}{k-1}} \vert \Gamma _{1} \vert + ( e\alpha ) ^{-\frac{k}{k-1}} ( K_{2} ) ^{ \frac{k}{k-1} ( p-1+\alpha ) } \Vert \nabla u \Vert _{2}^{\frac{k}{k-1} ( p-1+\alpha ) } \\ &\quad \leq H ( t ) +e ( p-1 ) ^{-\frac{k}{k-1}} \vert \Gamma _{1} \vert + ( e\alpha ) ^{- \frac{k}{k-1}} ( K_{2} ) ^{\frac{k}{k-1} ( p-1+ \alpha ) }H ( t ) ^{\frac{k}{k-1} ( p-1+\alpha ) }, \end{aligned}
(78)

where $$\vert x^{y} \ln x \vert \leq \frac{1}{e y}$$ for $$0< x<1$$ and $$x^{-y} \ln x \leq \frac{1}{e y}$$ for $$x \geq 1$$.

Inserting (78) into (77) gives

$$H^{\prime }(t)\leq H(t)+e(p-1)^{-\frac{k}{k-1}} \vert \Gamma _{1} \vert +(e\alpha )^{-\frac{k}{k-1}} ( K_{2} ) ^{ \frac{k}{k-1}(p-1+\alpha )}H(t)^{\frac{k}{k-1}(p-1+\alpha )}.$$
(79)

Using integration of (79) over t, we conclude

$${ \int}_{H(0)}^{H(t)} \frac{d\theta }{H(t)+e(p-1)^{-\frac{k}{k-1}} \vert \Gamma _{1} \vert +(e\alpha )^{-\frac{k}{k-1}} ( K_{2} ) ^{\frac{k}{k-1}(p-1+\alpha )}\theta ^{\frac{k}{k-1}(p-1+\alpha )}}.$$

It is easy to see that there is a time $$T_{1}$$ such that the solution goes to the infinity with $$\lim_{t \rightarrow T_{1}} H(t)=$$ ∞. Thus, we have a lower bound for $$T_{1}$$ given by

$$T_{1}\geq \frac{d\theta }{H(t)+e(p-1)^{-\frac{k}{k-1}} \vert \Gamma _{1} \vert +(e\alpha )^{-\frac{k}{k-1}} ( K_{2} ) ^{\frac{k}{k-1}(p-1+\alpha )}\theta ^{\frac{k}{k-1}(p-1+\alpha )}}.$$

This completed the proof. □

6 Conclusion

This work proves the existence of the result for a hyperbolic-type equation with logarithmic nonlinearity and dynamical boundary condition. This result is modern for these types of problems, and it can be generalized to many problems in the coming literature.

Data availability

There is no data associated to the current study.

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Irkıl, N., Mahdi, K., Pişkin, E. et al. On a logarithmic wave equation with nonlinear dynamical boundary conditions: local existence and blow-up. J Inequal Appl 2023, 159 (2023). https://doi.org/10.1186/s13660-023-03072-3