Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source

1.   Introduction

This paper is concerned with the existence and uniform decay rate estimates for the following initial boundary value problem:

where $\Omega$ is a bounded domain of $\mathrm{R}^{n}$ with $C^{4}$ boundary $\Gamma$. Let $\{\Gamma_{0},\Gamma_{1}\}$ be a partition of its boundary $\Gamma$ such that $\Gamma = \Gamma_{0}\cup\Gamma_{1}$, $\overline{\Gamma_{0}}\cap\overline{\Gamma_{1}} = \emptyset$ and $\Gamma_{0},\Gamma_{1}$ are positive measurable, endowed with the $(n-1)-$dimentional Lebesgue measure. Here, $\nu$ represents the unit outward normal to $\Gamma$, and $f_{i}$ $(i = 1,2)$ are given functions satisfying certain conditions to be specified later.

For the linear second order wave equations with nonlinear boundary feedback, there is an abounding literature about its initial boundary value problem. In [43], Zuazua studied the following second order wave equation

where $x_{0}$ is a fixed point in $\mathrm{R}^{n}$, and $m(x) = x-x_{0}$. When $f(y) = |y|^{p}$ on [0, 1] for some $p\geq1$, he proved that the energy decays exponentially if $p = 1$ and polynomially if $p>1$. In the later case, he gave that there exists a positive constant $C$ such that

When the nonlinear boundary velocity feedback $f(y)$ is weaker than any polynomial near the origin, for instance, $\forall\; y\in(0,1)$, $f(y) = e^{-1/y}$. Lasiecka and Tataru [20] showed that the energy of solutions decays with the following rate:

where $S(t)$ is the solutions (contraction semigroup) of the differential equation

and $q$ is closely related to the behavior of the feedback $f(y)$ near the origin. They were the first to consider that the energy decay rate estimates associated to the solutions of some differential equation and without assuming that the feedback has a polynomial behavior near the origin. Martinez [26] complemented Lasiecka and Tataru's work in [20] concerning the linear wave equation subject to nonlinear boundary feedback. He proved that the energy of problem (1.5) decays to zero with an explicit decay rate estimates. The process of the proof relies on the construction of some special weight functions and some nonlinear integral inequalities. The method presented in [26], gives us a variety of explicit decay rate estimates, although in some simple cases a direct application of the above method doesn't give us optimal decay rates. For instance, when $f(y) = y^{p}$, $p>1$, by the method of [26], the energy decay is given by $E(t)\leq{C}(1+t)^{-2/p}$, which is less good estimate than the estimate of (1.3). In spite of this, it is possible to obtain optimal decay rate estimates by this method for some other example, see [26] for details.

The linear second order wave equations subject to nonlinear boundary feedback and source terms have also been widely studied. For instance, Vitillaro [33] studied the following problem

He showed that the presence of the superlinear damping term $-|u_{t}|^{m-2}u_{t}$, when $2\leq{p}\leq{m}$, implies the global existence of solutions for arbitrary initial data, in opposition with the nonexistence phenomenon occurring when $m = 2<p$. Zhang and Hu [42] proved the asymptotic behavior of the solutions of problem (1.6), where the initial data is inside a stable set. The blow up phenomenon of the solutions occurs when the initial data is inside an unstable set. More results on the second order wave equations with nonlinear boundary source and damping terms, the reader can see [2,3,25] and papers cited therein.

It is worth mentioning that the potential well theory (stable or unstable sets) is a very important and popular way to study the qualitative properties of nonlinear evolution equations. This method was first introduced by Sattinger [30] to investigate the global existence of solutions for nonlinear hyperbolic equations. Hence, it has been widely used and extended by many authors to study different kinds of evolution equations, we refer the reader to see [6,7,30,34,35,36,38,39,40] and references therein.

Let us mention some known results about the second order wave equations with nonlinear internal damping and source terms

Geogev and Todorova [13] investigated the initial boundary value problem of equation (1.7), where $g(u_{t}) = |u_{t}|^{m-1}u_{t}$, $f(u) = |u|^{p-1}u$. They proved the existence of global solutions under the condition $1<p\leq{m}$. When $p\geq{m}>1$, they also obtained the finite time blow up of solutions for sufficient large initial data. Ikehata [15] studied the initial boundary value problem of equation (1.7), where $g(u_{t}) = \delta|u_{t}|^{m-1}u_{t}$ and $f(u) = |u|^{p-1}u$. He proved that $1\leq{m}<p<\infty$ if $n = 1,2$, and $1\leq{m}<p\leq\frac{n}{n-2}$ if $n\geq3$, the problem has a global solution for sufficiently small initial data. When $g(u_{t}) = au_{t}(1+|u_{t}|^{m-2})$, $f(u) = b|u|^{p-2}u$, Messaoudi [1,27] investigated the global existence and exponential decay behavior of solutions respectively.

For the second order wave equations with nonlinear internal source and boundary velocity feedback, Cavalcanti et al. [4] studied the following initial boundary value problem

They proved the existence of global solutions and uniform decay rate estimates of the energy provided that the nonlinear boundary feedback $f(u_{t})$ has not a polynomial growth near the origin by using the potential well method and the Galerkin approximation. When $f(u_{t}) = -\alpha(x)|u_{t}|^{m-2}u_{t}$ or $f(u_{t}) = -\alpha(x)(|u_{t}|^{m-2}u_{t}+|u_{t}|^{\mu-2}u_{t})$, $1\leq\mu\leq{m}$, and $\alpha(x)\in{L}^{\infty}(\Gamma_{1}), \alpha(x)\geq0$, Vitillaro [31] extended the potential well theory. He obtained the local existence, blow up and global existence results of solutions. More results on the initial boundary value problem for the wave equations with nonlinear internal source and boundary velocity feedback, we refer readers to see (Di and Shang [9], Feng and Li [11,12], Liu, Sun and Li [24]) and the papers cited therein.

There are some literature on the initial boundary value problem or Cauchy problem for the fourth order wave equations with source and damping terms in the interior of $\Omega$

For example, when $g(u_{t}) = a|u_{t}|^{m-2}u_{t}$, $f(u) = -q(x)u(x,t)$ with $q(x)>0$, Guesmia [14] investigated the initial boundary value problem of equation (1.9). He obtained a global existence and a regularity result and proved that the solutions decay exponentially if $g(y)$ behaves like a linear functions. For more results on the qualitative problem of the fourth order wave equations with interior source and damping terms, the reader is referred to see [8,37,41] and references therein.

When people studied the small transversal vibrations of a thin plate (Lagnese and Lions [17], Lagnese [18]) and the strong or uniform stabilization of different plate and beam models (Lasiecka [19], Puel and Tucsnak [28]), some nonlinear evolution equations with the main part $u_{tt}+\triangle^{2}{u} = 0$ and different nonlinear boundary feedbacks were obtained. For example, Komornik [16] studied the following evolutionary problem:

where $\mu\in(0,1)$, $l\in{C}^{1}(\Gamma_{1})$, and $f: \mathrm{R}\rightarrow\mathrm{R}$ is a non-decreasing, continuous function. The subscripts $\nu$ and $\tau$ stand for the normal and tangential derivatives to $\Gamma_{0}$ and $\Gamma_{1}$. He proved the global existence, regularity results and gave some stabilization properties for problem (1.10) by using the Multiplier method. It is worth mentioning that the Multiplier method has already been used by many authors for different reasons, we also refer to the related papers [5,16,21] about the Multiplier method.

Motivated by the above results, in the present work we study the initial boundary value problem of the fourth order wave equation with an internal nonlinear source $|u|^{\rho}u$, and nonlinear boundary velocity feedbacks $f_{1}(u_{\nu{t}})$, $f_{2}(u_{t})$. As far as we know, there is little information on the well-posedness and energy decay estimates for problem (1.1). Naturally, our attention of this paper is paid to the study of the related qualitative properties to problem (1.1). Here, when the boundary velocity feedbacks $f_{1}(u_{\nu{t}})$, $f_{2}(u_{t})$ have not the polynomial behaviour near the origin for wave equation supplemented with an interior source $|u|^{\rho}u$ acting in the domain, we first investigate the global existence, uniqueness of regular solutions and weak solutions by the combination of Galerkin approximation, potential well method and a special basis constructed. In addition, we also prove that the energy of problem (1.1) decays uniformly to zero, which is based on a weight function $\phi(t)$ constructed, Multiplier method and nonlinear integral inequality.

Our paper is organized as follows. In Section 2, we introduce some potential wells, basic definitions, important lemmas, and main results of this paper. In Section 3-4, we show the global existence and uniqueness of the regular solutions and weak solutions respectively. In the last Section, we investigate the explicit decay rate estimates of the energy.

2.   Preliminaries and main results

In order to state our results precisely, we first introduce some notations, basic definitions, important lemmas and some functional spaces.

Let $\Omega$ be a bounded domain of $\mathrm{R}^{n}$ with $C^{4}$ boundary $\Gamma$ and $x^{0}$ be a fix point in $\mathrm{R}^{n}$. We shall define

and introduce a partition of the boundary $\Gamma$ such that

Throughout this paper, the following inner products and norms are used for precise statement:

and the Hilbert space

Since $\Gamma_{0}$ has positive $(n-1)$ dimensional Lebesgue measure, by Poincaré inequality, we can endow $V$ with the equivalent norm $\|u\|_{V} = \|\triangle{u}\|_{2}$ (see [22]).

To obtain the results of this paper, let us consider the potential energy

and total energy

associated to the solutions of problem (1.1). We may define the (positive) number

which is also called the depth of the potential well. Moreover, the value $d$ is shown to be the Mountain pass level associated to the elliptic problem

Here, let $B_{1}>0$ be the optimal constant of Sobolev imbedding from $V$ into $L^{\rho+2}(\Omega)$, which satisfies the inequality $\|u\|_{\rho+2}\leq{B}_{1}\|\triangle{u}\|_{2}$, $\forall$ $u\in{V}$. From this inequality, we discover that

Furthermore, setting

and the function

We can easily see (the simple proof can be founded in [32]) that

where $\lambda_{1}$ is the absolute maximum point of function $f$.

Now, we will give some basic hypotheses to establish the main results of this paper.

(A1) Suppose that $0<\rho<\frac{4}{n-4}$, if $n\geq5$ and $\rho>0$, if $n = 1,2,3,4$. Then, we have the following Sobolev imbedding

(A2) Assumptions on the functions $f_{i}$ $(i = 1,2)$ : $f_{i}$: $\mathrm{R}\rightarrow\mathrm{R}$ are nondecreasing $C^{1}$ functions such that $f_{i}(0) = 0$. In addition, there exist some strictly increasing and odd functions $g_{i}$ of $C^{1}$ class on $[-1,1]$ satisfy

where $g_{i}^{-1}(s)$ denote the inverse functions of $g_{i}(s)$ and $C_{i1},C_{i2}$ are positive constants.

In order to obtain the global existence of regular solutions, we shall need the following additional hypotheses.

(A3) Assumptions on the initial data: let us consider

satisfying the compatibility conditions

Moreover, assume that

(A4) $E(0)<d$ and $\|\triangle{u}^{0}\|_{2}<\lambda_{1}$.

The next lemma will play an essential role for proving the global existence of regular (weak) solutions of problem (1.1).

Lemma 2.1. Suppose that $(A1)$, $(A2)$ and $(A4)$ hold. Let $u$ be a solution of problem (1.1), then for all $t\geq0$, $\|\triangle{u}(t)\|_{2}<\lambda_{1}$.

Proof. In view of (2.2), (2.6) and (2.7), we deduce that

where $f(\lambda) = \frac{1}{2}\lambda^{2}-K_{0}\lambda^{\rho+2},\; \; \lambda>0$, which is defined as (2.7). Of course, $f$ is increasing for $0<\lambda<\lambda_{1}$, decreasing for $\lambda>\lambda_{1}$, and $f(\lambda_{1}) = d$. From the definition of $f$, we also note that $f(\lambda)\rightarrow{+}\infty$ as $\lambda\rightarrow\infty$. Since $E(0)<d$, there exists $\lambda'_{2}<\lambda_{1}<\lambda_{2}$ such that $f(\lambda'_{2}) = f(\lambda_{2}) = E(0)$.

Multiplying the equation in (1.1) by $u_{t}(t)$, a direct computation gives that

By the hypotheses that $f_{i}$ are nondecreasing $C^{1}$ functions such that $f_{i}(0) = 0$, we know that $f_{i}(s)s>0$ for $s\neq0$. Hence, from the definition of $E(t)$, it follows that

So we have $E(t)\leq{E}(0)$ for all $t\geq0$. Denote $\lambda_{0} = \|\triangle{u}^{0}\|_{2}$, from the hypotheses $(A4)$ we have $\lambda_{0}<\lambda_{1}$. Furthermore, by (2.14), we have $f(\lambda_{0})\leq{E}(0)$, which together with $f$ is increasing in $[0,\lambda_{1})$ and $f(\lambda'_{2}) = E(0)$, it is easy to see that $\lambda_{0} = \|\triangle{u}^{0}\|_{2}<\lambda_{2}'$.

Next, we prove that $\|\triangle{u}(t)\|_{2}\leq\lambda'_{2}$ for all $t\geq0$. In deed, by contradiction, suppose that $\|\triangle{u}(t_{0})\|_{2}>\lambda'_{2}$ for some $t_{0}\geq0$. Using the continuity of $\|\triangle{u}(t)\|_{2}$, we also may suppose that $\|\triangle{u}(t_{0})\|_{2}<\lambda_{1}$. Thus, by (2.14) again, we see that

which contradicts (2.16). This completes the proof of Lemma 2.1.

The following two technical lemmas are very crucial to derive the asymptotic behavior of the energy to problem (1.1).

Lemma 2.2. Let $E:\; \mathrm{R}_{+}\rightarrow\mathrm{R}_{+}$ be a non-increasing function and $\phi:\mathrm{R}_{+}\rightarrow\mathrm{R}_{+}$ a strictly increasing function of $C^{1}$ class such that

Suppose that there exist $\sigma>0$, $\sigma'\geq0$ and $C>0$ such that

Then, there exists $C>0$ such that

Remark 2.1. Note that the above integral inequality was first introduced in Martinez [26], was used in Cavalcanti et al.[4] to prove the decay rate estimates of energy.

Lemma 2.3. There exists a strictly increasing function $\phi:\mathrm{R}_{+}\rightarrow\mathrm{R}_{+}$ of $C^{2}$ class on $(0,+\infty)$, and such that the following conditions hold

where the functions $g_{i}^{-1}(s)$ $(i = 1,2)$ were introduced in assumption $(A2)$.

Proof. These properties of the function $\phi$ are closely related to the behaviors of $f_{i}$ $(i = 1,2)$ near 0. We will present the construction method of a special weight function $\phi$ in Section 5.

Now, we are ready to state the main results of this paper.

Theorem 2.1 (Existence and uniqueness of regular solutions). Let the assumptions $(A1)-(A4)$ hold, then the problem (1.1) possesses a unique regular strong solution $u$ satisfying

for all $t\geq0$. Further, the following energy identity holds

where the total energy $E(t)$ has been defined by (2.2).

Theorem 2.2 (Existence and uniqueness of weak solutions). Given $\{u^{0}, u^{1}\}\in{V}\cap{L}^{2}(\Omega)$. Assume that the hypotheses $(A1)$, $(A2)$ and $(A4)$ hold, then the problem (1.1) possesses a unique weak solution satisfying

for all $t>0$. Besides, the weak solution has the same energy identity given as (2.24).

Theorem 2.3 (Uniform decay rates of energy). Assume that the hypotheses $(A1)-(A4)$ hold. Let $u$ be a solution to problem (1.1) with the properties listed in Theorem 2.1. Then, the energy of problem (1.1) has the following decay rate

where the function $G(y) = y\frac{g_{1}(y)g_{2}(y)}{g_{1}(y)+g_{2}(y)}$ and the constant $C$ only depending on the initial data $E(1)$ in a continuous way.

Remark 2.2. By a direct calculation, we can show that the $G(y) = y\frac{g_{1}(y)g_{2}(y)}{g_{1}(y)+g_{2}(y)}$ is an increasing function.

Remark 2.3. we also extend the decay rate estimate of regular solutions to the weak solutions of problem (1.1) by using the standard arguments of density.

3.   Existence, uniqueness of regular solutions

In this section, we study the global existence and uniqueness of regular solutions of problem $(1.1)$ by using the combination of the Galerkin approximation, potential well method and a special basis constructed.

The proof of Theorem 2.1 is divided into five steps.

Proof. Step 1. Galerkin approximation.

The main idea is to use the Galerkin's method. To do this, let us take a basis $\{w_{j}^{*}\}$ to $V$. We construct a special basis $\{w_{j}\}$ from basis $\{w_{j}^{*}\}$ which are associated with problem (1.1).

If $u^{0}$, $u^{1}$ are linearly independent, we take $w_{1} = u^{0}$, $w_{2} = u^{1}$, and $w_{i}$, $i\geq3$ of $\{w_{j}^{*}\}$, which are chosen to be linearly independent with $u^{0}$, $u^{1}$. If $u^{0}$, $u^{1}$ are linearly dependent, we define $w_{1} = u^{0}$, and $w_{i}$, $i\geq2$ of $\{w_{j}^{*}\}$, which are chosen to be linearly independent with $u^{0}$. Thus, we represent by $V_{m}$ a subspace of $\{w_{j}\}$ generated by $[w_{1},\cdots,w_{m}]$.

Next, we construct an approximate solution of problem (1.1) by

According to Galerkin's method, these coefficients $d_{m}^{j}(t)$ need to satisfy the following initial value problem of the nonlinear ordinary differential equation

Note that we can solve system (3.2) by Picard's iteration method. In fact, the ordinary differential equation (3.2) has a local solution on the interval $[0,T_{m})$. The extension of these solutions to the whole interval $[0,+\infty)$ is a consequence of a priori estimate which we are going to prove below.

Step 2. The first estimate.

Replacing $w_{j}$ by $u^{m}_{t}$ in (3.2), a direct computation gives that

which implies that $E_{m}(t)$ is a decreasing function.

Combining problem (3.2) and assumption $(A4)$, we obtain that

Taking Lemma 2.1 into account, we conclude that $\|\triangle{u}^{m}(t)\|_{2}<\lambda_{1}$, for all $t\geq0$. Returning to the approximate problem, we deduce

Considering assumption $(A1)$, we have Sobolev inequality $\|u^{m}(t)\|_{\rho+2}\leq{B_{1}} \|\triangle{u}^{m}(t)\|_{2}$, which together with above inequality, a simple calculation reveals that

Step 3. The second estimate.

Multiplying (3.2) by $d{''}_{m}^{j}(0)$, summing for $j = 1,2,\cdots\cdots$, and considering $t = 0$, then we have

Using the generalized Green Theorem, it follows that

By Hölder inequality and the compatibility condition $(A3)$, we discover that

Differentiating equation in (3.2) with respect to $t$, and substituting $w_{j}$ by $u^{m}_{tt}$, we deduce that

We will give the estimate of $K_{1} = (\rho+1)\int_{\Omega}|u^{m}|^{\rho}|u^{m}_{t}||u^{m}_{tt}|dx$. From now on, we will denote by $C$ various positive constants which may be different at different occurrences.

In view of the generalized Hölder inequality ($\frac{\rho}{2(\rho+1)}+\frac{1}{2(\rho+1)}+\frac{1}{2} = 1$), Sobolev imbedding $V\hookrightarrow{L}^{2(\rho+1)}(\Omega)$ and Lemma 2.1, we conclude that

where the constant $C$ are positive constants independent of $m$ and $t$. By (3.9) and (3.10), it is inferred that

Integrating the above inequality over $(0,t)$, and taking (3.8) into account, we get that

The Gronwall Lemma guarantees that

From the inequality (3.13) and Trace Theorem [10], we also obtain the following estimate

where the constant $C>0$ is independent of $m$ and $t$. Furthermore, taking assumption $(A2)$ into account, we know that if $|u^{m}_{t}(t)|>1$, then $|f_{2}(u_{t}^{m}(t))|\leq{C_{22}}|u^{m}_{t}(t)|$. If $|u^{m}_{t}(t)|\leq1$, we obtain from the continuity of the function $f_{2}$ that $|f_{2}(u_{t}^{m}(t))|\leq{C}$. Thereby, we obtain that

Using analogous arguments, from the assumption $(A2)$ and (3.14), we also obtain that

Step 4. Global existence.

From the above estimates, we can show that there exists a subsequences of $\{u^{m}\}$ which from now on will be also denoted by $\{u^{m}\}$ and function $u:\; \Omega\times[0,T]$ such that

Since $V\hookrightarrow{L}^{2(\rho+1)}(\Omega)\hookrightarrow{L}^{2}(\Omega)$ is compact, thanks to Aubin-Lions Theorem [38, Chapter 1], we have that

Consequently, making use of Lion's Lemma [38,Lemma 1.3,Chapter 1], it follows that

In addition, we also obtain

Therefore, (3.19)-(3.27) permit us to pass to the limit in equation (3.2). Since $\{w_{j}\}$ is a basis of $V$, then for all $T>0$, for all $d(t)\in{D}(0,T)$ and for all $w\in{V}$, we have

Taking into account $w\in{D}(\Omega)$ and (3.28), we deduce that

Utilizing the convergences of (3.19) and (3.24), there appear the relations that $u_{tt}\in{L}^{\infty}(0,T;L^{2}(\Omega))$ and $|u|^{\rho}u\in{L}^{\infty}(0,T;L^{2}(\Omega))$. Hence, we deduce that $\triangle^{2}u\in{L}^{\infty}(0,T;L^{2}(\Omega))$ and

Combining (3.19) and (3.26), it is easy to see that the approximate solutions $\{u^{m}\}$ possess the following property

for all $w\in{V}$, which implies that

Taking (3.28)-(3.30) into account, and making use of generalized Green formula, we discover that

Since $\chi_{1},\chi_{2}\in{L}^{\infty}(0,T;L^{2}(\Gamma_{1}))$, we deduce that

Next, we need to prove that

In deed, replacing $w_{j}$ by $u^{m}$ in equation (3.2), and integrating the obtained expression over $(0,T)$, it is inferred that

In view of the first and second estimates, Sobolev imbedding, Poincaré inequality, and Trace Theorem [10], it follows that

which implies that

Making use of the Aubin-Lions Theorem [23,Chapter 1] again, we have that

Then, from the convergences (3.19), (3.24), (3.26), (3.27) and (3.37), we can pass to the limit in equation (3.34) to obtain

Combining (3.29), (3.32), (3.39) and the generalized Green formula, it is found that

which implies that

Now, in view of (3.26), (3.27), (3.38), and using the standard Lebesgue control-convergent Theorem, we obtain that

Utilizing the non-decreasing monotonicity of functions $f_{i}$ $(i = 1,2)$, it follows that

for all $\psi\in{L}^{2}(\Gamma_{1})$. Then, from the inequalities (3.43), (3.44), we discover that

and then passing to the limit as $m\rightarrow\infty$,

In order to prove (3.33) from (3.47) and (3.48), we use the semi-continuous [23,Chapter 2]. Let $\psi = u_{\nu{t}}-\lambda\varphi$, $\forall\; \varphi\in{L}^{2}(\Gamma_{1})$ and $\lambda\geq0$, then we have

and

Pass to the limit as $\lambda\rightarrow0$ gives that

In a similar way, let $\psi = u_{\nu{t}}-\lambda\varphi$, $\lambda\leq0$ and $\forall\; \varphi\in{L}^{2}(\Gamma_{1})$, we obtain

From (3.50) and (3.51), we see that

Using the analogous arguments, taking $\psi = u_{\nu{t}}-\lambda\varphi$, and $\forall\; \varphi\in{L}^{2}(\Gamma_{1})$, we also get from (3.48) that

which implies that

Thus, we obtain that $u$ is a global regular solutions of problem (1.1).

Step 5. Uniqueness.

Let $u$, $\widetilde{u}$ be two solutions of problem (1.1). Then, $y = u-\widetilde{u}$ satisfies

for all $w\in{V}$. Replacing $w$ by $y_{t}$ in the above identity, and noting that $f_{i}$ $(i = 1,2)$ are monotone functions, it follows that

Using the Hölder inequality, Sobolev imbedding $V\hookrightarrow{L}^{2(\rho+1)}(\Omega)$ and taking the first estimate into account, we thereby deduce that

Then, apply the Gronwall Lemma yields that $\|y_{t}(t)\|_{2}^{2} = \|\triangle{y}(t)\|_{2}^{2} = 0$. This completes the proof of Theorem 2.1.

4.   Existence, uniqueness of weak solutions

Our attention in this section is turned to the existence, uniqueness of weak solutions for problem (1.1). Applying the standard density argument, we extend the existence, uniqueness results of regular solutions to the weak solutions.

Proof. The main idea of this proof is the density method. We will divided it into four steps.

Step 1. Galerkin approximation.

We start to approximate the initial data $u^{0}$ and $u^{1}$ with more regular data $u_{\mu}^{0}$ and $u_{\mu}^{1}$, respectively. Indeed, let us assume that

such that

Hence, we choose

where $D(\triangle^{2}) = \{u\in{V}\cap{H}^{4}(\Omega);\; u_{\nu\nu\nu} = u_{\nu\nu} = 0\; \; \text{on}\; \; \Gamma_{1}\}$ such that

Thus, it is easy to see that $\{u_{\mu}^{0}, u_{\mu}^{1}\}$ satisfies the compatibility conditions

Moreover, using the continuity of functionals $\|\triangle{u}\|_{2}$, $E(u)$, we have

where $E_{\mu}(0) = E(u_{\mu}^{0})$. Therefore, for sufficiently large $\mu\geq\mu_{0}$, we get

Thus, for each $\mu\geq\mu_{0}$, let $u^{\mu}$ be the solutions of problem (1.1) with the initial date $\{u_{\mu}^{0}, u_{\mu}^{1}\}$, which satisfies all the conditions of Theorem 2.1, so we obtain

and verifies

Step 2. Energy estimates and global existence.

Applying the analogous arguments used to prove the first estimate of the above section, we deduce that there exist constants $C$ (various positive constants $C$ may be different at different occurrences) which are independent of $\mu$ and $t\in[0,T]$, such that

Let us define $y^{\mu,\sigma}(t) = u^{\mu}(t)-u^{\sigma}(t)$, $\mu,\sigma\in{N}$. From the monotonicity of functions $f_{i}$, $(i = 1,2)$, it follows that

which together with the Hölder inequality, Sobelev imbedding from $V\hookrightarrow{L}^{2(\rho+1)}(\Omega)$ and (4.8) gives that

Then, the Gronwall Lemma reveals that

where the constant $C>0$ is independent of $\mu,\sigma\in{N}$.

Consequently, the estimates (4.11) and (4.3) permit us to obtain a subsequences of $u^{\mu}$ which from now on will be also denoted by $u^{\mu}$ and function $u$ such that for all $T>0$,

On the other hand, from (4.8) and (4.12), we also obtain

Considering the above convergences, making use of the arguments of compactness and generalized Green formula, we deduce that

Combining (4.3), (4.12), (4.13) and (4.17), it follows that $\triangle^{2}u\in{C}(0,T;H^{-2}(\Omega))$, $|u|^{\rho}u\in{C}(0,T;L^{2}(\Omega))$, and

From the identity (4.18), making use of the Bochner's integral in $H^{-2}(\Omega)$, it follows that

Defining $Z(t) = \int_{0}^{t}u(s)ds$, so we obtain from (4.19) that

Furthermore, thanks to (4.12), (4.13) and $(A1)$, we discover that

By the first equation of problem (1.1), we note that $\triangle^{2}Z(t)\in{C}(0,T;L^{2}(\Omega))$, which implies that

where $\mathcal{H}(\Omega) = \{u\in{H}^{2}(\Omega);\; \triangle^{2}u\in{L}^{2}(\Omega)\}$. Together with the definition of $Z(t)$, we have that

Similarly, if we define $Z^{\mu}(t) = \int_{0}^{t}u^{\mu}(s)ds$, using the same arguments as (4.12), (4.13) and (4.17), we obtain that

In view of (4.22)-(4.24), making use of Lion's Lemma [38, Lemma 1.3, Chapter 1] yields that

Combining (4.15), (4.16) and the above convergences, it is inferred that

On the other hand, from the convergences of (4.13) and (4.17), we know that $u_{t}(t)\in{C}(0,T;L^{2}(\Omega))$ and $|u|^{\rho}u\in{L}^{\infty}(0,T;L^{2}(\Omega))$. By the Sobolev embedding relations ${C}(0,T;L^{2}(\Omega))\hookrightarrow {L}^{2}(0,T;L^{2}(\Omega))$ and ${L}^{\infty}(0,T;L^{2}(\Omega))\hookrightarrow {H}^{-1}(0,T;L^{2}(\Omega))$, if follows that $u_{t}(t)\in{L}^{2}(0,T;L^{2}(\Omega))$ and $|u|^{\rho}u\in{H}^{-1}(0,T;L^{2}(\Omega))$. Hence, it is easy to see that $u_{tt}(t)\in{H}^{-1}(0,T;L^{2}(\Omega))$ and

Utilizing the above identity, the generalized Green formula and (4.25), it is found that

which along with Trace Theorem, Sobolev imbedding $L^{2}(0,T;V)\hookrightarrow{L}^{2}(0,T; H^{\frac{3}{2}}(\Gamma_{1}))$ $\hookrightarrow{L}^{2}(0,T;L^{\frac{1}{2}}(\Gamma_{1}))$ and Hölder inequality leads to

for all $v\in{H}_{0}^{1}(0,T;V)$. Thus, the term $\triangle^{2}u$ possess a continuous extension to the space ${L}^{2}(0,T;V')$ such that

Next, our goal is to show that

In deed, multiplying the first equation in (4.7) by $u_{t}^{\mu}$ and integrating over $\Omega$, we have

Integrate (4.35) over $(0,t)$ leads to

Considering the convergences (4.3), (4.12) and (4.13), we deduce that

On the other hand, we assume that $u$ is a weak solution to the problem

Adapting the ideas of Lasiecka and Tataru [2, Proposition 2.1], Komornik [33, Theorem 7.9] or Lions [38, Lemma 6.1], we obtain that the weak solutions $u$ satisfy energy identity

which along with (4.37) yields to

Taking (4.14)-(4.16) into account, we get that

and

By the analogous arguments which have been used in the proof's process of regular solutions. we also obtain from (4.36) and (4.37) that $\chi_{1} = f_{1}(u_{\nu{t}})$, and $\chi_{2} = f_{2}(u_{t})$. Thus, we prove that there exists the global weak solutions $u$ satisfying

with $\|\triangle{u(t)}\|_{2}<\lambda_{1}$ for all $t\geq0$.

Step 3. Uniqueness.

Finally, we will use the standard energy estimate to get the uniqueness of weak solutions. Let $u$ and $\widetilde{u}$ be the solutions of problem (4.43), then $y = u-\widetilde{u}$ satisfies

Making use of the same procedure to prove (4.39), we have the energy identity

which together with the Hölder inequality, assumptions $(A2)$ and (4.8) leads to

Employing the Gronwall Lemma, we get that $\|y_{t}(t)\|_{2}^{2}+\|\triangle{y}(t)\|_{2}^{2} = 0$, which implies the uniqueness of weak solutions. This completes the proof of Theorem 2.2.

5.   Uniform decay rates of solutions

The focus of the development in this section is the decay rate estimates of the energy to problem (1.1). The proofs are based on the construction of a special weight function $\phi$, nonlinear integral inequality and the Multiplier method.

First, by the virtue of Theorem 2.1, it is known that the solution $u$ of problem (1.1) possesses the some properties listed in Theorem 2.1 and Theorem 2.2. Thus, we can apply the following energy identity

Taking into account that $f_{i}(s)s>0$ if $s\neq0$, we see that $E(t)$ is a non-increasing function. Moreover, the weight function $\phi$ appeared in Lemma 2.3 (construction method of $\phi$ will be presented in the sequel) will play key role in the proof of energy decay rate estimates.

Now, let us multiply the equation in (1.1) by $E\phi'Mu$, where the function $Mu$ is defined by

Then, considering $0\leq{S}<T<+\infty$ and applying the generalized Green formula, we deduce that

Estimate of $I_{1} = 2\int_{S}^{T}E\phi'\int_{\Omega}u_{tt}(m\cdot\nabla{u})dxdt$.

Applying integration by parts and Gauss Theorem, it follows that

Estimate of $I_{2} = 2\int_{S}^{T}E\phi'\int_{\Omega}\triangle{u}\cdot\triangle(m\cdot\nabla{u})dxdt$.

The application of Gauss Theorem gives that

Estimate of $I_{3} = (n-1)\int_{S}^{T}E\phi'\int_{\Omega}u_{tt}udxdt$.

By the integration by parts again, we also obtain that

Inserting (5.4)-(5.6) into (5.3), noting that $u_{\nu\nu} = -f_{1}(u_{\nu{t}}),\; u_{\nu\nu\nu} = f_{2}(u_{t})$ on $\Gamma_{1}$ and $\nabla{u} = u_{\nu}\cdot\nu$ on $\Gamma_{0}$, it follows that

Using the definition of energy $E(t)$ and the identity (5.7), we obtain that

Next, we shall estimate the last two terms of the right hand side of the above identity (5.8).

Estimate of $D_{1} = \left[n-1+\frac{2}{\rho+2}\right]\int_{S}^{T}E\phi'\int_{\Omega}|u|^{\rho+2}dxdt$.

Taking into account that $\frac{1}{p} = \frac{\alpha}{2}+\frac{1-\alpha}{q}$, $\alpha\in[0,1]$, then by the interpolation inequality of $L^{p}(\Omega)$ spaces, $\|s\|_{p}\leq\|s\|_{2}^{\alpha}\|s\|_{q}^{1-\alpha}$ with $p = \rho+2$, $q = 2(\rho+1)$ and $\alpha = \frac{1}{\rho+2}$, we deduce that

Setting $h = n-1+\frac{2}{\rho+2}$, by Poincaré inequality, Sobolev embedding from $V$ into $L^{2(\rho+1)}(\Omega)$ and Young inequality, we obtain that

for all $\varepsilon>0$ and $B_{2} = \left(\frac{2(\rho+2)}{\rho}\right)^{\rho+1}E(0)^{\rho}$. Combining (2.8) and (2.14), a direct computation gives that

which implies that

Furthermore, replace (5.12) in (5.10) gives that

From (5.13), we obtain that

Estimate of $D_{2} = 2\int_{S}^{T}E\phi'\int_{\Omega}|u|^{\rho}u(m\cdot\nabla{u})dxdt$.

By the Hölder inequality and Poincaré inequality, we have that

Taking into account that $0<\rho<\frac{4}{n-4}$, if $n>4$, and $0<s<\frac{2n}{n-4}-2(\rho+1)$, and considering the interpolation inequality $\|s\|_{p}\leq\|s\|_{2}^{\alpha}\|s\|_{q}^{1-\alpha}$ with $p = 2(\rho+1)$, $q = 2(\rho+1)+s$, we discover that

where $\alpha\in(0,1)$ is given by $\alpha = 1+\frac{s}{(\rho+1)[2-2(\rho+1)-s]}$, which implies that

Applying Poincaré inequality and Sobolev embedding from $V\hookrightarrow{L}^{2(\rho+1)+s}(\Omega)$ $\left(2(\rho+1)+s<\frac{2n}{n-4}\right)$, then we have

Combining (5.15) and (5.18), we conclude that

From the Young inequality,

for all $\varepsilon>0$. Let us take $p = \frac{2}{(1-\alpha)(\rho+1)}$ and $p' = \frac{2}{2-(1-\alpha)(\rho+1)}$, then we have

where

Combining (5.19) and (5.21), we have that

Therefore, in view of (5.8), (5.14) and (5.22), when $m\cdot\nu\leq0$ on $\Gamma_{0}$ and $\varepsilon$ small enough, we can conclude that there exists $\delta_{1},\delta_{2}>0$ such that

In order to estimate the last term of (5.23), let us give the following lemma.

Lemma 5.1. Under the hypotheses of Theorem 2.1. Let $u$ be a solution to problem (1.1). Then for $T>T_{0}$, where $T_{0}$ is sufficiently large, we have

for all $0\leq{S}<T<+\infty$.

Proof. We shall argue by contradiction. Suppose that (5.24) is not verified. Let ${u_{k}}$ be a sequence of solutions to problem (1.1) such that

while the total energy $E_{k}(0)$ with initial data $\{u^{k}(0),u^{k}_{t}(0)\}$ remains uniformly bounded in $k$, that is, there exists $\widetilde{M}>0$ such that $E_{k}(0)<\widetilde{M}$.

Since $E_{k}(0)<\widetilde{M}$, by the non-increasing property of $E_{k}(t)$, we have $E_{k}(t)<\widetilde{M}$. Hence, there exists a subsequence of the sequence $\{u_{k}\}$, still denoted by $\{u_{k}\}$, which satisfies

Applying the similar methods used to prove (3.18) and (3.25), we have that

Notice that the Aubin-Lions type compactness gives us

In what follows, we will apply the ideas contained in Lasiecka and Tataru [20] or Cavalcanti et al [4] to our context.

Case (ⅰ). Let us consider that $u\neq0$. By (5.30), it follows that

Since the sequence $\{|u^{k}|^{\rho}u^{k}\}$ is bounded in ${L}^{\infty}(0,T;L^{2}(\Omega))$, together with (5.32) and Lion's Lemma [23,Lemma 1.3,Chapter 1], we have

Taking into account that the Poincaré inequality and the boundedness of $E_{k}(t)$, it is found that

where $C$ is a positive constant independent of $k$ and $t$. Thus, we can deduce that the term $\int_{S}^{T}\phi'\int_{\Omega}|\nabla{u}^{k}|^{2}dxdt$ is bounded. Therefore, we have from (5.25) that

as $k\rightarrow\infty$. Especially, (5.35) implies that

Since $\phi(t)$ is concave, it follows that $\phi'(t)\geq\phi'(T)$, $t\in[S,T]$, for any $T>0$, we also get

Thus, combining (5.36) and (5.37), it follows that

Considering that $S$ is chosen in the interval $[0,T]$, so we write

Therefore, we conclude

In a similar way, we also conclude that

Taking $k\rightarrow+\infty$ in the equation, we get for $u$

and for $u_{t} = v$,

Note that $u\in{L}^{\infty}(0,T;V)$ implies $u\in{L}^{\frac{4}{n-4}}(\Omega)$ and $(\rho+1)|u|^{\rho}\in{L}^{\infty}(0,T;L^{n}(\Omega))$. Applying the standard uniqueness results of [16,see Chapter 6] or the uniqueness results of [29] to our context, we conclude that $v = 0$, which means $u_{t} = 0$, for $T$ suitably large.

Hence, the equation (5.42) reduce to the elliptic equation

Multiplying the above elliptic equation by $u$, we have

which implies that $J(u) = \frac{\rho}{2(\rho+2)}\|\triangle u\|_{2}^{2}$. But according to (5.11), it follows that

for all $u\neq0$. This is a contradiction.

Case (ⅱ). Let us assume that $u\equiv0$. Setting

which implies

Besides,

By the similar argument as (5.46), we deduce that

which along with (5.49) yields that

Also,

Furthermore, when $u\equiv0$, we deduce that $c_{k}\rightarrow0$ as $k\rightarrow+\infty$.

On the other hand, considering the energy identity,

and multiplying this identity by $E_{k}(t)$, then we obtain

Integrating (5.54) with respect to $t$ from $S$ to $T$, we discover that

In view of (5.54) and (5.55), we deduce that

Replacing $Mu^{k} = 2(m\cdot\nabla{u}^{k})+(n-1)u^{k}$ in inequality (5.23), we obtain that

Estimate of $G_{1} = -2\int_{S}^{T}E_{k}\phi'\int_{\Gamma_{1}}f_{2}(u^{k}_{t})(m\cdot{u_{\nu}^{k}})d\Gamma{d}t$.

Using Young inequality and a direct calculation gives that

for all $\eta>0$.

Estimate of $G_{2} = -2\int_{S}^{T}E_{k}\phi'\int_{\Gamma_{1}}f_{1}(u^{k}_{\nu{t}})(m\cdot{u}_{\nu}^{k})_{\nu}d\Gamma{d}t$.

Applying integration by parts and the Young inequality, it follows that

Estimate of $G_{3} = -2E_{k}\phi'\int_{\Omega}u^{k}_{t}(m\cdot\nabla{u}^{k})dx$.

Considering Young inequality and Poincaré inequality, we obtain from the definition of $E_{k}(t)$ that

where $L$ is a positive constant which verifies $|\phi'(t)|\leq|\phi'(0)| = L$, $\forall\; t\geq0$. Therefore, we have

Estimate of $G_{4} = -(n-1)\left[E_{k}\phi'\int_{\Omega}u^{k}_{t}u^{k}dx\right]_{S}^{T}$.

Analogously, considering the same procedure used to prove (5.61), we also get that

Estimate of $G_{5} = \int_{S}^{T}(E_{k}'\phi'+E_{k}\phi'')\int_{\Omega}u^{k}_{t}Mu^{k}dxdt$.

By Young inequality and Poincar$\acute{e}$ inequality, a simple computation reveals that

We thereby conclude that

Estimate of $G_{6} = -(n-1)\int_{S}^{T}E_{k}\phi'\int_{\Gamma_{1}}f_{2}(u^{k}_{t})u^{k}d\Gamma{d}t$.

Using Young inequality, there appears the relation

for any $\gamma>0$. Taking into account that the continuity of the linear trace operator $\mathfrak{B}$: $V\hookrightarrow{H}^{1}(\Gamma_{1})\hookrightarrow{L}^{2}(\Gamma_{1})$, there exist two positive constants $\xi_{1},\xi_{2}$ such that

for all $u\in{V}$. Hence, we deduce that

Estimate of $G_{7} = -(n-1)\int_{S}^{T}E_{k}\phi'\int_{\Gamma_{1}}f_{1}(u^{k}_{\nu{t}})u_{\nu}^{k}d\Gamma{d}t$.

Analogously, we obtain that

Since $m\cdot\nu$ are sufficiently smooth and $\Gamma_{1}$ is compact, there exists $\delta>0$ such that $m\cdot\nu\geq\delta>0$ for all $x\in\Gamma_{1}$. Consequently, inserting the estimates $(G_{1})-(G_{7})$ into (5.57), we conclude that

Taking $\eta,\gamma$ small enough such that $\delta_{1}-2C\gamma>0$ and $1-\frac{\eta}{\delta}-\frac{2C\eta}{\delta}>0$, then we have

where ${C}_{i}$, $i = 1,\cdots,6$ are positive constants. Combining (5.56) and (5.70), it is found that

Furthermore, considering that $\phi'(t)$ is a non-increasing function, it is inferred that

Since $\phi(t)\rightarrow+\infty$ as $t\rightarrow+\infty$, for a large $T$, it is noted that $\phi(T)-\phi(S)-{C}_{3}-{C}_{4}\phi'(S)>0$. Thus, using Poincar$\acute{e}$ inequality again, we deduce that

Dividing both sides of the last inequality by $\int_{S}^{T}\phi'\int_{\Omega}|\nabla{u}^{k}|^{2}dxdt$, then for very $t\in[S,T]$, where $0\leq{S}<T<+\infty$, we have that

By (5.25), we know that

therefore, there exists $\widetilde{N}>0$ such that

for all $t\in[S,T]$, $0\leq{S}<T<+\infty$. Combining (5.51) and (5.75), we have that

which implies

for all $t\in[S,T]$, $0\leq{S}<T<+\infty$.

Hence, there exists a subsequence of the sequence $\{\widetilde{u}_{k}\}$, still denoted by $\{\widetilde{u}_{k}\}$, which satisfies

In addition, $\widetilde{u}^{k}$ also satisfies

From (5.74), we see that

Since

we thereby have

which implies

Making use of the same procedure used to prove (5.85), we deduce that

Further, there appear the relation

Considering that $|y|^{\rho}$ is a continuous in $R$, so we define $\widetilde{M}_{\varepsilon} = \sup_{|y|\leq\varepsilon}|y|^{\rho}$. Therefore, we obtain

Combining (5.77) and hypotheses $(A1)$, we deduce from the (5.88) that

Then, taking $\varepsilon\rightarrow0$ and $k\rightarrow+\infty$, we get that

From what has been discussed above, passing to the limit in (5.81) as $k\rightarrow+\infty$, we have

Differentiating (5.91) with respect to $t$ and taking $v = \widetilde{u}_{t}$, we conclude that

Applying the standard uniqueness results of [16](see Chapter 6) or the uniqueness results of [29] to our context again, it comes that $v = 0$, that is $\widetilde{u}_{t} = 0$. Returning to the equation (5.91), we obtain

Multiplying the above problem by $\widetilde{u}$, we see that

which implies that $\widetilde{u} = 0$. But from the (5.48) and (5.80), we conclude that $\widetilde{u}\neq0$. This is a contraction. This completes the proof of Lemma 5.1.

On the basis of Lemma 5.1, we are now in positive to give the straightforward proof of Theorem 2.3.

Proof of Theorem 2.3. Inserting the results of Lemma 5.1 into (5.23) and then using the similar calculation as (5.57) to (5.70), we have that

Analysis of $J_{1} = \int_{S}^{T}\phi'\int_{\Gamma_{1}}|u_{\nu{t}}|^{2}d\Gamma{d}t$.

For every $t\geq1$, let us define the following partition of $\Gamma_{1}$:

where $\Gamma_{1,1},\Gamma_{1,2},\Gamma_{1,3}$ depend on $t\in\mathrm{R}^{+}$ and ${h}_{1}(t) = g_{1}^{-1}(\phi'(t))$ is a decreasing positive function and satisfies ${h}_{1}(t)\rightarrow0$, as $t\rightarrow+\infty$.

Estimate of $\Gamma_{1,3}\;$: we note that ${h}_{1}(1) = 0$ $\Leftrightarrow$ $g_{1}^{-1}(\phi'(1)) = 0$ $\Leftrightarrow$ $\phi'(1) = g_{1}(0) = 0$. But, if $\phi'(1) = 0$, we have for $t\geq0$ that $\phi'(t)\leq\phi'(1) = 0$. Consequently, $\phi'(t) = 0$, $\forall\; {t}\geq1$, which contradicts the fact that $\phi$ is a strictly increasing function. Thus, we have ${h}_{1}(1)>0$.

If ${h}_{1}(1)>1$, we obtain from the hypotheses $(A2)$ that $|f_{1}(u_{\nu{t}})|\geq{C}_{11}|u_{\nu{t}}|$.

If ${h}_{1}(1)\leq1$, we observe that the function $F:\; y\rightarrow\frac{f_{1}(y)}{y}$ is a positive and continuous on $[-1,-{h}_{1}(1)]$$\cup$ $[{h}_{1}(1),1]$, which implies that there exists a constant $\beta_{1}>0$ such that $\frac{f_{1}(y)}{y}\geq\beta_{1}$, that means $|f_{1}(u_{\nu{t}})|\geq\beta_{1}|u_{\nu{t}}|$.

So we conclude that $|u_{\nu{t}}|\leq\frac{1}{d_{0}}|f_{1}(u_{\nu{t}})|$, where $d_{0} = \min\{{C}_{11},\beta_{1}\}$. Therefore, we have

Estimate of $\Gamma_{1,2}\;$: considering that $g_{1}$ is an increasing function, it follows that $\phi'(t) = g_{1}(h_{1}(t))$$\leq{g}_{1}(|u_{\nu{t}}|)$$\leq|{g}_{1}(u_{\nu{t}})|$.

If ${h}_{1}(1)<1$, we deduce that $|u_{\nu{t}}|<1$. By the hypotheses $(A2)$, we get that $|g_{1}(u_{\nu{t}})|\leq|f_{1}(u_{\nu{t}})|\leq|g_{1}^{-1}(u_{\nu{t}})|$. Thus, it follows that $|u_{\nu{t}}|^{2}|g_{1}(u_{\nu{t}})|\leq|u_{\nu{t}}|^{2}|f_{1}(u_{\nu{t}})|\leq|u_{\nu{t}}||f_{1}(u_{\nu{t}})|$.

If ${h}_{1}(1)\geq1$ and $|u_{\nu{t}}|\in[1,{h}_{1}(1)]$, we have that $-{h}_{1}(1)\leq{u}_{\nu{t}}\leq{h}_{1}(1)$. Since $g_{1}$ is an increasing and odd function, it is found that $|g_{1}(u_{\nu{t}})|\leq|g_{1}({h}_{1}(1))|$. Thus, taking into account that $C_{11}|u_{\nu{t}}|\leq|f_{1}(u_{\nu{t}})|$, we have

which implies that

Hence, we discover that $|u_{\nu{t}}|^{2}|g_{1}(u_{\nu{t}})|\leq{d}_{1}u_{\nu{t}}f_{1}(u_{\nu{t}})$, where $d_{1} = \min\{1,\frac{g_{1}({h}_{1}(1))}{C_{11}}\}$. Furthermore, taking into account that $|\phi'(t)| = |g_{1}(h_{1}(t))|\leq|{g}_{1}(u_{\nu{t}})|$, we obtain that

Estimate of $\Gamma_{1,1}\;$: thanks to the definition of this part of the boundary, we have that

Therefore, in view of (5.97)-(5.100), there appears the relation

where $L_{1},L_{2}$ are positive constants.

Analysis of $J_{2} = \int_{S}^{T}\phi'\int_{\Gamma_{1}}|u_{{t}}|^{2}d\Gamma{d}t$.

For every $t\geq1$, let us define the following partition of $\Gamma_{1}$:

where $\Gamma_{1,4},\Gamma_{1,5},\Gamma_{1,6}$ depend on $t\in\mathrm{R}^{+}$ and ${h}_{2}(t) = g_{2}^{-1}(\phi'(t))$ is a decreasing positive function which satisfies ${h}_{2}(t)\rightarrow0$, as $t\rightarrow+\infty$.

By a straightforward adaptation of the above result (5.101), we also obtain that

where $L_{3},L_{4}$ are positive constants.

Analysis of $J_{3} = \int_{S}^{T}\phi'\int_{\Gamma_{1}}|f_{1}(u_{\nu{t}})|^{2}d\Gamma{d}t$.

For every $t\geq1$, let us define the following partition of $\Gamma_{1}$:

where $\Gamma_{1,7},\Gamma_{1,8},\Gamma_{1,9}$ depend on $t\in\mathrm{R}^{+}$.

Estimate of $\Gamma_{1,9}\;$: if $\phi'(1) = 0$, we have that for all $t\geq1$, $\phi'(t)\leq\phi'(1) = 0$. Consequently, $\phi'(t) = 0,\; \forall\; t\geq1$, which contradicts the fact that $\phi(t)$ is a strictly increasing function. Then, $\phi'(1)>0$.

If $\phi'(1)>1$, we obtain from the hypotheses $(A2)$ that $|f_{1}(u_{\nu{t}})|\leq{C}_{12}|u_{\nu{t}}|$.

If $\phi'(1)\leq1$, we observe that the function $F:\; y\rightarrow\frac{f_{1}(y)}{y}$ is a positive and continuous on $[-1,-\phi'(1)]$$\cup$ $[\phi'(1),1]$ which implies that there exists a constant $\beta_{2}>0$ such that $\frac{f_{1}(y)}{y}\leq\beta_{2}$, that means $|f_{1}(u_{\nu{t}})|\leq\beta_{2}|u_{\nu{t}}|$.

We conclude that $|f_{1}(u_{\nu{t}})|\leq{d}_{3}|u_{\nu{t}}|$, where $d_{3} = \max\{{C}_{12},\beta_{2}\}$. Therefore, we have that

Estimate of $\Gamma_{1,8}\;$: considering the monotonicity of $f_{1}$, $f_{1}(\phi'(t))<f_{1}(|u_{\nu{t}}|)\leq{f}_{1}(\phi'(1))$, and the boundary conditions of this part, we discover that

Estimate of $\Gamma_{1,7}\;$: if $\phi'(1)\leq1$, we have from the hypotheses $(A2)$ that $|f_{1}(u_{\nu{t}})|\leq|g_{1}^{-1}(u_{\nu{t}})|\leq|{g}_{1}^{-1}(\phi'(t))|$. Then,

If $\phi'(1)>1$, then $|u_{\nu{t}}|\in[1,\phi'(t)]$. From the hypotheses $(A2)$, we obtain $|f_{1}(u_{\nu{t}})|\leq{C}_{12}|u_{\nu{t}}|$, and

Therefore, combining (5.105)-(5.108), we have

where $L_{5},L_{6}$ are positive constants.

Analysis of $J_{4} = \int_{S}^{T}\phi'\int_{\Gamma_{1}}|f_{2}(u_{{t}})|^{2}d\Gamma{d}t$.

For every $t\geq1$, let us define the following partition of $\Gamma_{1}$:

where $\Gamma_{1,10},\Gamma_{1,11},\Gamma_{1,12}$ depend on $t\in\mathrm{R}^{+}$.

Using the analogous arguments as (5.109), we obtain

where $L_{7},L_{8}$ are positive constants.

Inserting (5.101), (5.103), (5.109) and (5.111) into inequality (5.95), it follows that

Now assume that $\phi$ satisfies the following additional properties:

These properties are closely related to the behavior of $g_{i}$ $(i = 1,2)$ near 0 and the decay rate of $\phi'$ at infinity. Thus, we deduce from (5.112) and (5.113) that there exists positive constants $C$ such that

Next, the main problem is to find a strictly increasing function, which satisfies the following conditions:

We consider, without loss of generality, that $\phi(1) = 1$. From this fact, observing that $\phi^{-1}(t)\rightarrow+\infty\; \; \text{as}\; t\rightarrow+\infty$ and taking into account the change of variable $\tau = \phi(t)$, one have that

Let us define the auxiliary functions $\psi_{1},\psi_{2}$ on $[1,\infty)$ by

Then $\psi_{1},\psi_{2}$ are strictly increasing functions of $C^{2}$ on $[1,\infty)$ and satisfy

and

By a direct computation, we can show that $\psi_{1}''(t),\psi_{2}''(t)\geq0$ which imply that $\psi'_{1},\psi'_{2}$ are non-increasing functions and $\psi_{1},\psi_{2}$ are convex. Furthermore, let $\psi = \psi_{1}+\psi_{2}$, then it is easy to verify that $\psi^{-1}(t)$ is concave on $[1,\infty)$. Take two derivatives of the expression $\psi(\psi^{-1}(t)) = t$, we deduce that

That is why we define $\phi$ on $[1,\infty)$ by

Thus $\phi$ is a strictly increasing concave function of class $C^{2}$ on $[1,\infty)$, which satisfies all the assumptions we made in our computation. In addition, we deduce that

Note that $\phi(1) = 1$, because $\psi(1) = \psi_{1}(1)+\psi_{2}(1) = 1$, and $\phi'(1) = \frac{g_{1}(1)g_{2}(1)}{g_{1}(1)+g_{2}(1)}\leq\frac{1}{2}$, so it is easy to extend $\phi$ on $[0,\infty)$ such that it remains a concave and strictly increasing nonnegative function on $[0,\infty)$. Thus, we have explicitly constructed a function $\phi$ that satisfies the all the conditions of Lemma 2.3.

Furthermore, we deduce from the (5.114) that

We define $F(t) = E(t+1)$ and $\Phi(t) = \phi(t+1)-1$ on $[0,+\infty)$, then (5.126) implies that

Noting that the function $F(t)$ and $\Phi(t)$ satisfy all the assumptions of Lemma 2.2 with $\sigma = \sigma' = 1$, so we obtain a decay rate estimate:

By the method of variable substitution, we also obtain

where $C$ is depending on $E(1)$ in a continuous way.

Finally, it remains to estimate the growth of $\phi$. Setting $\tau_{0}$ such that

Using the monotonicity of $g_{i}$ $i = 1,2$, we have

Hence, we have that

which implies that

Taking into account that

so we have

where the function $G(y) = y\frac{g_{1}(y)g_{2}(y)}{g_{1}(y)+g_{2}(y)}$. Thus, we thereby conclude that

which completes the proof of Theorem 2.3.

Acknowledgments

The authors wish to express their gratitude to the Editors and Referees for giving a number of valuable comments and helpful suggestions, which led us to improve the presentation of original manuscript significantly. This work was initiated while Di was visiting Department of Mathematics, University of Texas at Arlington as a Postdoctoral Researcher during the year 2018-2020, who would like to thank the department for its warm hospitality and support.