Abstract
Motivated by the recent known results as regards the existence and exponential decay of solutions for wave equations, this paper is devoted to the study of an N-dimensional nonlinear wave equation with a nonlocal boundary condition. We first state two local existence theorems. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. The main tools are the Faedo-Galerkin method and the Lyapunov method.
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1 Introduction
In this paper, we consider the following initial-boundary value problem:
where Ω is a bounded domain in \(\mathbb{R}^{N}\) with a smooth boundary ∂Ω, ν is the unit outward normal on ∂Ω; \(a=\pm1\), K, λ, p are given constants, and \(u_{0}\), \(u_{1}\), f, g, h are given functions satisfying conditions specified later.
The wave equation
with different boundary conditions, has been extensively studied by many authors, for example, we refer to [1–25] and the references given therein. In these works, many interesting results about the existence, regularity and the asymptotic behavior of solutions were obtained. In [3], Beilin investigated the existence and uniqueness of a generalized solution for the following wave equation with an integral nonlocal condition:
where Ω is a bounded domain in \(\mathbb{R}^{N}\) with a smooth boundary, ν is the unit outward normal on ∂Ω, f, \(u_{0}\), \(u_{1}\), \(k(x,\xi,\tau)\) are given functions. Nonlocal conditions come up when values of the function on the boundary is connected to values inside the domain. There are various type of nonlocal boundary conditions of integral form for hyperbolic, parabolic or elliptic equations, introduced in [3]. In [4], the following problem was considered:
where \(f(u)=-b|u|^{p-2}u\), \(g(u_{t})=a(1+|u_{t}|^{m-2})u_{t}\), \(a, b>0\), \(m, p>2\), and Ω is a bounded domain of \(\mathbb{R}^{N}\), with a smooth boundary ∂Ω. Benaissa and Messaoudi showed that for suitably chosen initial data, (1.6) possesses a global weak solution, which decays exponentially even if \(m>2\). The proof of the global existence is based on the use of the potential well theory.
As [4], Messaoudi [10] also showed the problem (1.6), with \(f(u)=b|u|^{p-2}u\), \(b>0\) has a unique global solution with energy decaying exponentially for any initial data \(( u_{0},u_{1} ) \in H^{1}(\Omega)\times L^{2}(\Omega)\). So if \(f(u)=b|u|^{p-2}u\), and \(g(u_{t})=|u_{t}|^{m-2}u_{t}\), Nakao [16] showed that (1.6) has a unique global weak solution if \(0\leq p-2\leq\frac{2}{N-2}\), \(N\geq 3\), and a global unique strong solution if \(p-2>\frac{2}{N-2}\), \(N\geq3\) (of course if \(N=1\) or \(N=2\) then the only requirement is \(p\geq2\)). On the other hand, in both cases it has been shown that the energy of the solution decays algebraically if \(m>2\) and decays exponentially if \(m=2\). Also as [4], Nakao and Ono [17] extended this result to the Cauchy problem,
where \(g(u_{t})\) behaves like \(|u_{t}|^{m-2}u_{t}\), \(f(u)\) behaves like \(-|u|^{p-2}u\) and the initial data \((u_{0},u_{1}) \) is small enough in \(H^{1}(\Omega)\times L^{2}(\Omega)\). Later on, Ono [19] studied the global existence and the decay properties of smooth solutions to the Cauchy problem related to (1.6), for \(f(u)\equiv0\) and gave sharp decay estimates of the solution without any restrictions on the data size \(( u_{0},u_{1} ) \).
In [21], Munoz-Rivera and Andrade dealt with the global existence and exponential decay of solutions of the nonlinear one-dimensional wave equation with a viscoelastic boundary condition. In [22–24], Santos also studied the asymptotic behavior of solutions to a coupled system of wave equations having integral convolutions as memory terms. The main results show that the solutions of that system decay uniformly in time, with rates depending on the rate of decay of the kernel of the convolutions.
In [25], the global existence and regularity of weak solutions for the linear wave equation
with the initial conditions as in (1.3) and two-point boundary conditions. The exponential decay of solutions was also given there by using Lyapunov method.
The works introduced as above lead to the study of the existence and exponential decay of solutions for the problem (1.1)-(1.3). This paper consists of three sections. The preliminaries are presented and two existence results with \(a=1 \) are done in Section 2. The decay of the solution with respect to \(a=1\), \(g=0\), \(K>0\), \(\lambda>0\), and \(2< p\leq \frac{2N-2}{N-2}\), \(N\geq3 \) is established in Section 3. The proofs of the existences are based on the Faedo-Galerkin method for strong solutions and standard arguments of density for weak solutions. Because this problem is solved in an N-dimensional domain, it causes technical difficulties, so we need the relations between the norms as in Lemmas 2.1-2.3 below. To obtain the exponential decay, we use the multiplier technique combined with a suitable Lyapunov functional in the form \(\mathcal{L}(t)=E(t)+\delta \psi (t)\), where
\(\delta>0\) is chosen sufficiently small, which allows us to show that if
and if the initial energy \(E(0)\), f, h given are small enough, then the energy \(E(t) \) of the solution decays to zero exponentially when t goes to infinity.
We end the paper with a remark about a situation where \(a=-1\), precisely we consider (1.1) in the form
With some suitable conditions for f, h, g, we obtain a unique global solution for (1.2)-(1.3) and (1.9), with energy decaying exponentially as \(t\rightarrow+\infty\), without any restrictions on the data size \((u_{0},u_{1})\) as in [19].
2 Preliminaries and existence results
In this paper, \(\Omega\subset \mathbb{R}^{N}\) is an open and bounded set with a smooth boundary ∂Ω and the usual function spaces \(C^{m} ( \overline{\Omega} ) \), \(W^{m,p}=W^{m,p} ( \Omega ) \), \(L^{p}=W^{0,p} ( \Omega ) \), \(H^{m}=W^{m,2} ( \Omega ) \), \(1\leq p\leq\infty\), \(m=0,1,\ldots \) are used. Let \(\langle\cdot,\cdot\rangle\) be either the scalar product in \(L^{2}\) or the dual pairing of a continuous linear functional and an element of a function space. The notation \(\Vert \cdot \Vert \) stands for the norm in \(L^{2}\) and we denote by \(\Vert \cdot \Vert _{X}\) the norm in the Banach space X. We call \(X^{\prime}\) the dual space of X. We denote by \(L^{p}(0,T;X)\), \(1\leq p\leq\infty \), the Banach space of the real functions \(u:(0,T)\rightarrow X\) measurable, such that
and
Let \(u(t)\), \(u^{\prime}(t)=u_{t}(t)\), \(u^{\prime\prime }(t)=u_{tt}(t)\), \(\nabla u(t)\), \(\Delta u(t)\) denote \(u(x,t)\), \(\frac{\partial u}{\partial t}(x,t)\), \(\frac{\partial^{2}u}{\partial t^{2}}(x,t)\), \((\frac{\partial u}{\partial x_{1}}(x,t), \ldots, \frac{\partial u}{\partial x_{N}}(x,t))\), \(\sum_{i=1}^{N}\frac{\partial^{2}u}{\partial x_{i}^{2}}(x,t)\), respectively.
On \(H^{1}\) we shall use the following norm: \(\Vert v\Vert _{H^{1}}= ( \Vert v\Vert ^{2}+\Vert \nabla v \Vert ^{2} ) ^{1/2}\).
In cases \(N=1\) or \(N=2\), by the continuity and compactness of the injections \(H^{1}(\Omega)\hookrightarrow C^{0}(\overline{\Omega})\) with \(N=1\) or \(H^{1}(\Omega)\hookrightarrow L^{q}(\Omega)\) with \(N=2\), it is not difficult to study problem (1.1)-(1.3). On the other hand, it is obvious that the problem considered with \(a=1\) is more difficult than the one with \(a=-1\),so in what follows we only consider problem (1.1)-(1.3) with \(N\geq3\), \(a=1\). A remark in the end of this paper will give a note in the case \(a=-1\).
First, we recall the following results, see [26].
Lemma 2.1
Let \(\Omega\subset \mathbb{R}^{N}\) be an open and bounded set of class \(C^{1}\). Then the embedding \(H^{1}\hookrightarrow L^{q}\), is continuous if \(1\leq q\leq2^{\ast}\) and compact if \(1\leq q<2^{\ast}\), where \(2^{\ast}=\frac{2N}{N-2}\), \(N\geq3\).
Lemma 2.2
Let \(\Omega\subset \mathbb{R}^{N}\) be an open and bounded set with a smooth boundary ∂Ω. Then
where \(\gamma_{\Omega}\) is a positive constant depending only on the domain Ω.
The proofs below also require the following lemma.
Lemma 2.3
Let \(\Omega\subset \mathbb{R}^{N}\) be an open and bounded set with a smooth boundary ∂Ω. Let \(2\leq p\leq\frac{2N-2}{N-2}\), \(N\geq3\). Then there exists a constant \(D_{p}>0\) depending on p, N and Ω such that
for all \(u, v\in H^{1}\).
Proof
(i) We have
with \(W=\vert u\vert +\vert v\vert \).
Hence, by Hölder’s inequality we have
Note that \(H^{1}\hookrightarrow L^{q}\), \(1\leq q\leq2^{\ast}=\frac {2N}{N-2}\), \(N\geq3\), and \(\Vert v\Vert _{L^{q}}\leq C_{q}\Vert v\Vert _{H^{1}}\), \(\forall v\in H^{1}\), \(1\leq q\leq2^{\ast}\).
Choose \(\alpha=\frac{2^{\ast}}{2}=\frac{N}{N-2}\), we have \(\alpha ^{\prime }=\frac{\alpha}{\alpha-1}=\frac{\frac{N}{N-2}}{\frac{N}{N-2}-1}=\frac {N}{2}\), and
By the condition \(2\leq p\leq\frac{2N-2}{N-2}=2+\frac{2}{N-2}\), \(N\geq3\) is equivalent to
so we consider two cases as follows.
Case 1. \(1\leq(2p-4)\alpha^{\prime}\leq2^{\ast}=\frac {2N}{N-2}\):
Case 2. \(0\leq\beta\equiv(2p-4)\alpha^{\prime}<1\leq 2^{\ast}=\frac{2N}{N-2}\):
Consequently, in both cases we get
Hence
Similarly (ii) is proved.
The proof of Lemma 2.3 is complete. □
Next, we state two local existence theorems. We make the following assumptions:
- (\(\mathrm{A}_{0}\)):
-
\(2< p\leq\frac{2N-2}{N-2}\), \(N\geq3\),
- (\(\mathrm{B}_{0}\)):
-
\(K, \lambda\in \mathbb{R}\),
- (\(\mathrm{A}_{1}\)):
-
\(f, f^{\prime}\in L^{1}(0,T;L^{2})\),
- (\(\mathrm{A}_{2}\)):
-
\(h\in L^{2}(0,T;L^{2} ( \partial\Omega\times\Omega ) )\), \(h^{\prime}, h^{\prime\prime}\in L^{2}(0,T;L^{2} ( \partial\Omega\times\Omega ) )\),
- (\(\mathrm{A}_{3}\)):
-
\(g\in L^{2} ( \partial\Omega\times\Omega ) \), \(g^{\prime}, g^{\prime\prime}\in L^{2} ( \partial\Omega\times \Omega ) \),
- (\(\mathrm{A}_{1}^{\prime}\)):
-
\(f\in L^{2}(Q_{T})\),
- (\(\mathrm{A}_{2}^{\prime}\)):
-
\(h\in L^{2}(0,T;L^{2} ( \partial\Omega\times \Omega ) )\), \(h^{\prime}\in L^{2}(0,T;L^{2} ( \partial\Omega \times\Omega ) )\),
- (\(\mathrm{A}_{3}^{\prime}\)):
-
\(g\in L^{2}(0,T;L^{2}(\partial\Omega))\), \(g^{\prime }\in L^{2}(0,T;L^{2}(\partial\Omega))\).
Then we have the following theorem as regards the existence of a ‘strong solution’.
Theorem 2.4
Suppose that (\(\mathrm{A}_{0}\)), (\(\mathrm{B}_{0}\)), (\(\mathrm{A}_{1}\))-(\(\mathrm{A}_{3}\)) hold and the initial data \(( u_{0},u_{1} ) \in H^{2}\times H^{1}\) satisfies the compatibility condition
Then problem (1.1)-(1.3) has a unique local solution
for \(T_{\ast}>0\) small enough.
Remark 2.1
The regularity obtained by (2.13) shows that problem (1.1)-(1.3) has a unique strong solution
With less regular initial data, we obtain the following theorem as regards the existence of a weak solution.
Theorem 2.5
Let (\(\mathrm{A}_{0}\)), (\(\mathrm{B}_{0}\)), (\(\mathrm{A}_{1}^{\prime }\))-(\(\mathrm{A}_{3}^{\prime}\)) hold and \(( u_{0},u_{1} ) \in H^{1}\times L^{2}\).
Then problem (1.1)-(1.3) has a unique local solution
for \(T_{\ast}>0\) small enough.
Proof of Theorem 2.4
Let \(\{w_{j}\}\) be a denumerable base of \(H^{2}\). Under the assumptions of Theorem 2.4, using the Faedo-Galerkin approximation and Lemmas 2.1-2.3, we find the approximate solution of problem (1.1)-(1.3) in the form
where the coefficient functions \(c_{mj}\) satisfy the system of ordinary differential equations
From the assumptions of Theorem 2.4, system (2.17) has a solution \(u_{m}\) on an interval \([0,T_{m}]\subset[0,T]\). The following estimates allow one to take \(T_{m}=T_{\ast}\) for all m, consisting of two key estimates.
In the first key estimate, we put \(S_{m}(t)=\Vert u_{m}^{\prime }(t)\Vert ^{2}+\Vert \nabla u_{m}(t)\Vert ^{2}\), it implies from (2.17) that
By Lemmas 2.1-2.3 and the following inequalities:
and
with computing explicitly, all terms in the right-hand side of (2.18) are estimated, in which the following estimates are worthy of note:
since \(1\leq2\leq2p-2\leq2^{\ast}\), and \(H^{1}(\Omega )\hookrightarrow L^{2p-2}(\Omega)\), we have
it leads to
Combining estimations of all terms and choosing \(\beta=\frac{1}{4}\), we obtain after some rearrangements
where \(C_{T}\) always indicates a constant depending on T.
Then, by solving a nonlinear Volterra integral inequality (2.29) (based on the methods in [27]), the following lemma is proved.
Lemma 2.6
There exists a constant \(T_{\ast}>0\) depending on T (independent of m) such that
where \(C_{T}\) is a constant depending only on T.
By Lemma 2.6, we can take a constant \(T_{m}=T_{\ast}\) for all m.
In the second key estimate, we put \(X_{m}(t)=\Vert u_{m}^{\prime \prime }(t)\Vert ^{2}+\Vert \nabla u_{m}^{\prime}(t)\Vert ^{2}\),and it follows from (2.17) that
Letting \(t\rightarrow0_{+}\) in equation (2.17)1, multiplying the result by \(c_{mj}^{\prime\prime}(0)\), and using the compatibility (2.12), we get
This implies that
where \(\overline{X}_{0}\) is a constant depending only on p, K, λ, \(u_{0}\), \(u_{1}\), f.
Also note the following estimations:
From
by Lemma 2.3(ii), it gives
where
Combining estimations and choosing \(\beta=\frac{1}{2}\), we obtain after some rearrangements
where \(C_{T}\) always indicates a constant depending on T, and
By Gronwall’s lemma, we deduce from (2.40) that
It verifies the existence of a subsequence of \(\{u_{m}\}\), denoted by the same symbol, such that
By the compactness lemma of Lions ([28], p.57), we can deduce from (2.43) the existence of a subsequence still denoted by \(\{u_{m}\}\), such that
By means of the continuity of the function \(t\longmapsto|t|^{p-2}t\), we have
On the other hand
Using the Lions lemma ([28], Lemma 1.3, p.12), it follows from (2.45) and (2.46) that
Passing to the limit in (2.17) by (2.43), (2.44), and (2.47), we have u satisfying the problem
On the other hand, we have from (2.43), (2.48)1
Thus \(u\in L^{\infty}(0,T_{\ast};H^{2})\) and the proof of existence is complete. The uniqueness of a weak solution is proved as follows.
Let \(u_{1}\), \(u_{2}\) be two weak solutions of problem (1.1)-(1.3), such that
Then \(u=u_{1}-u_{2}\) satisfy the variational problem
We take \(v=u^{\prime}=u_{1}^{\prime}-u_{2}^{\prime} \) in (2.51) and integrating with respect to t, we obtain
where
By (2.53) and the following inequalities:
we estimate the following integrals in the right-hand side of (2.52):
By Lemma 2.3(i), we have
where \(M_{1}=\Vert u_{1}\Vert _{L^{\infty} ( 0,T_{\ast };H^{1} ) }+\Vert u_{2}\Vert _{L^{\infty} ( 0,T_{\ast };H^{1} ) }\). Hence
Combining (2.52), (2.56)-(2.58), (2.60) and choosing \(\beta=\frac{1}{2}\), we obtain
By Gronwall’s lemma, it follows from (2.61) that \(\sigma\equiv0\), i.e., \(u_{1}\equiv u_{2}\). Theorem 2.4 is proved completely. □
Proof of Theorem 2.5
In order to prove this theorem, we use standard arguments of density.
First, we note that \(W^{1}(0,T;L^{2}(\partial\Omega))=\{g\in L^{2}(0,T;L^{2}(\partial\Omega)):g^{\prime}\in L^{2}(0,T;L^{2}(\partial \Omega))\}\) is a Hilbert space with respect to the scalar product (see [27]):
Furthermore, we also have the embedding \(W^{1}(0,T;L^{2}(\partial\Omega ))\hookrightarrow C^{0}([0,T];L^{2}(\partial\Omega))\) is continuous and
for all \(g\in W^{1}(0,T;L^{2}(\partial\Omega))\), where \(\gamma _{T}=\sqrt{\frac{1}{2T}+\sqrt{4+\frac{1}{4T^{2}}}}\) (see the Appendix).
Similarly, \(W^{1}(0,T;L^{2} ( \partial\Omega\times\Omega ) )=\{h\in L^{2}(0,T;L^{2} ( \partial\Omega\times\Omega ) ):h^{\prime}\in L^{2}(0,T;L^{2} ( \partial\Omega\times\Omega ) )\}\) is a Hilbert space with respect to the scalar product
and the embedding \(W^{1}(0,T;L^{2} ( \partial\Omega\times\Omega ) )\hookrightarrow C^{0}([0,T];L^{2} ( \partial\Omega\times \Omega ) )\) is continuous and
for all \(h\in W^{1}(0,T;L^{2} ( \partial\Omega\times\Omega ) )\), where \(\gamma_{T}=\sqrt{\frac{1}{2T}+\sqrt{4+\frac{1}{4T^{2}}}}\) (see the Appendix).
Consider \(( u_{0},u_{1},f,g,h ) \in H^{1}\times L^{2}\times L^{2}(Q_{T})\times W^{1}(0,T;L^{2} ( \partial\Omega ) )\times W^{1}(0,T;L^{2} ( \partial\Omega\times\Omega ) )\).
Let the sequence \(\{ ( u_{0m},u_{1m},f_{m},g_{m},h_{m} ) \} \subset H^{2}\times H^{1}\times C_{0}^{\infty} ( \overline{Q}_{T} ) \times C_{0}^{\infty} ( \partial\Omega\times\overline{\Omega} ) \times C_{0}^{\infty} ( \partial\Omega\times\overline{\Omega }\times [0,T] ) \), such that
So \(\{ ( u_{0m},u_{1m} ) \}\) satisfy, for all \(m\in \mathbb{N}\), the compatibility condition
Then, for each \(m\in \mathbb{N}\), there exists a unique function \(u_{m}\) under the conditions of Theorem 2.4. Thus, we can verify
and
By the same arguments used to obtain the above estimates, we get
\(\forall t\in[0,T_{\ast}]\), where \(C_{T}\) always indicates a constant depending on T as above.
On the other hand, we put \(w_{m, l}=u_{m}-u_{l}\), \(f_{m, l}=f_{m}-f_{l}\), \(h_{m, l}=h_{m}-h_{l}\), \(g_{m,l}=g_{m}-g_{l}\), \(h_{m, l}(x,y,0)=\bar{h}_{m, l}^{(0)}(x,y)\), \(g_{m, l}(x,0)=\bar{g}_{m, l}^{(0)}(x)\), from (2.70), it follows that
We take \(v=w_{m,l}=u_{m}-u_{l}\), in (2.73) and integrating with respect to t, we get
where
After all terms of \(S_{m,l}(t)\) are estimated, in which we note the two main estimations \(Z_{1}\), \(Z_{8} \) as follows:
this result combined with (2.66)-(2.68) shows that
On the other hand
by Lemma 2.3(i), we get
We obtain
with
as \(m, l\rightarrow+\infty\). By Gronwall’s lemma, it follows from (2.81) that
Thus, convergence of the sequence \(\{ ( u_{0m},u_{1m},f_{m},g_{m},h_{m} ) \}\) implies the convergence to zero as \(m, l\rightarrow+\infty\) of the term on the right-hand side of (2.83). Therefore, we get
On the other hand, from (2.72), we get the existence of a subsequence of \(\{u_{m}\}\), still also so denoted, such that
By the compactness lemma of Lions ([28], p.57), we can deduce from (2.85) the existence of a subsequence, still denoted by \(\{u_{m}\}\), such that
Similarly, by (2.72), it follows from (2.86) that
Passing to the limit in (2.70) by (2.84)-(2.87), we have u satisfying the problem
Next, the uniqueness of a weak solution is obtained by using the well-known regularization procedure due to Lions. Theorem 2.5 is proved completely. □
Remark 2.2
In the case \(1< p\leq2\), \(f\in L^{2}(Q_{T})\), \(g\in W^{1}(0,T;L^{2} ( \partial\Omega ) )\), \(h\in W^{1}(0,T;L^{2} ( \partial\Omega\times\Omega ) )\), and \(( u_{0},u_{1} ) \in H^{1}\times L^{2}\), the integral inequality (2.29) leads to the following global estimation:
Then, by applying a similar argument to the proof of Theorem 2.4, we can obtain a global weak solution u of problem (1.1)-(1.3) satisfying
However, in the case \(1< p<2\), we do not imply that a weak solution obtained here belongs to \(C ( [0,T];H^{1} ) \cap C^{1} ( [0,T];L^{2} ) \). Furthermore, the uniqueness of a weak solution is also not asserted.
3 Exponential decay
In this section, we study the exponentially decay of solutions of problem (1.1)-(1.3) corresponding to \(a=1\), \(g=0\), \(K>0\), \(\lambda >0\), and \(2< p\leq\frac{2N-2}{N-2}\). For this purpose, we make the following assumptions:
- (\(\mathrm{A}_{1}^{\prime\prime}\)):
-
\(f\in L^{2} ( 0,\infty;L^{2} ) =L^{2}(Q_{\infty})\), \(Q_{\infty}=\Omega\times\mathbb{R}_{+}\), such that \(\Vert f(t)\Vert \leq Ce^{-\gamma _{0}t}\), for all \(t\geq0\), with \(C>0\), \(\gamma_{0}>0\) are given constants,
- (\(\mathrm{A}_{2}^{\prime\prime}\)):
-
\(h\in L^{\infty}(0,\infty;L^{2} ( \partial\Omega\times\Omega ) )\cap L^{2} ( \mathbb{R} _{+}\times\partial\Omega\times\Omega ) \), \(h^{\prime\prime}\in L^{\infty}(0,\infty;L^{2} ( \partial\Omega\times\Omega ) )\cap L^{1}(0,\infty;L^{2} ( \partial\Omega\times\Omega ) )\),
- (\(\mathrm{A}_{3}^{\prime\prime}\)):
-
\(g=0\).
Let \(K>0\), on \(H^{1}\) we shall use the following norm:
Then we have the following lemma.
Lemma 3.1
On \(H^{1}\), two norms \(\Vert v\Vert _{1}\), \(\Vert v\Vert _{H^{1}}\) are equivalent and
where \(C_{0}=\frac{1}{\sqrt{\min\{1,K\}}}\).
The proof of this lemma is simple, we omit the details.
We construct the following Lyapunov functional:
where \(\delta>0\) is chosen later and
Put
we rewrite
Then we have the following theorem.
Theorem 3.2
Assume that (\(\mathrm{A}_{1}^{\prime\prime}\))-(\(\mathrm{A}_{3}^{\prime\prime}\)) hold. Let \(I(0)>0\) and the initial energy \(E(0)\) satisfy
where
\(\bar{\gamma}_{\Omega}=\gamma_{\Omega}C_{0}\) and \(C_{p}\) is a constant satisfying the inequality \(\Vert v\Vert_{L^{p}}\leq C_{p}\Vert v\Vert _{H^{1}}\), for all \(v\in H^{1}\).
Then, for \(E ( 0 ) \), \(\Vert f\Vert _{L^{\infty }(0,\infty;L^{2})}\), \(\Vert h\Vert _{L^{\infty}(0,\infty ;L^{2} ( \partial\Omega\times\Omega ) )}\), \(\Vert h^{\prime} \Vert _{L^{\infty}(0,\infty;L^{2} ( \partial \Omega \times\Omega ) )}\) sufficiently small, there exist positive constants C, γ such that
Proof of Theorem 3.2
At first, we state and prove Lemmas 3.3-3.6 as follows.
Lemma 3.3
The energy functional \(E(t)\) satisfies
Proof
Multiplying (1.1) by \(u^{\prime}(x,t)\) and integrating over \([0,1]\), we get
We have
By Lemmas 2.1, 2.2, 3.1, we obtain
Combining (3.11)-(3.14), (3.10) follows. Lemma 3.3 is proved completely. □
Lemma 3.4
Suppose that (\(\mathrm{A}_{1}^{\prime\prime}\))-(\(\mathrm{A}_{3}^{\prime\prime}\)) hold. Then, if we have \(I(0)>0\) and
then \(I(t)>0\), \(\forall t\geq0\).
Proof
By the continuity of \(I(t)\) and \(I(0)>0\), there exists \(T_{1}>0\) such that
this implies
It follows from (3.6), (3.17) that
Equation (3.10) leads to
Integrating with respect to t, we obtain
where \(\bar{h}(t)\) is as in (3.8).
Combining (3.18), (3.20), and using the Gronwall lemma, we have
and
Hence, it follows from (3.7), (3.22) that
Therefore, \(I(t)>0\), \(\forall t\in[0,T_{1}]\).
Now, we put \(T_{\infty}=\sup \{ T>0:I(u(t))>0, \forall t\in [0,T] \} \). If \(T_{\infty}<+\infty\) then, by the continuity of \(I(t)\), we have \(I(T_{\infty})\geq0\). By the same arguments as in the above part, we can deduce that there exists \(T_{2}>T_{\infty}\) such that \(I(t)>0\), \(\forall t\in[0,T_{2}]\). Hence, we conclude that \(I(t)>0\), \(\forall t\geq0\).
Lemma 3.4 is proved completely. □
Lemma 3.5
Let \(I(0)>0\) and (3.7) hold. Then there exist the positive constants \(\beta_{1}\), \(\beta_{2}\) such that
for δ is sufficiently small.
Proof
A simple computation gives
From the following inequalities:
we deduce from (3.25) that
where we choose
with δ being small enough, \(0<\delta<\min \{ \frac {1}{2}, \frac{\frac{1}{2}-\frac{1}{p}}{\frac{1}{2}C_{0}^{2} ( \lambda+\frac{1}{2} ) } \} \).
Similarly, we can prove that
where
Lemma 3.5 is proved completely. □
Lemma 3.6
Let \(I(0)>0\) and (3.7) hold. Then the functional \(\psi(t)\) defined by (3.3) satisfies
for all \(\varepsilon_{1}>0\).
Proof
By multiplying (1.1) by \(u(x,t)\) and integrating over \([0,1]\), we obtain
Note that
hence, Lemma 3.6 is proved by using some estimates. □
Now, we prove Theorem 3.2.
It follows from (3.1), (3.10), and (3.31) that
for all \(\delta, \varepsilon_{1}>0\), where
Let δ, \(\varepsilon_{1}\) satisfy
Then, for δ small enough such that \(0<\delta<\frac{\lambda}{2}\) and if h satisfy
we deduce from (3.34), (3.36), and (3.37) that there exists a constant \(\gamma>0\) such that
where
Combining (3.38) and (3.39), we get (3.9). Theorem 3.2 is proved completely. □
Remark
We consider the following problem:
With the suitable conditions for K, λ, p, \(u_{0}\), \(u_{1}\), f, g, h, we prove that problem (3.40) has a unique global solution \(u(t)\) with energy decaying exponentially as \(t\rightarrow+\infty\), without the initial data \(( u_{0},u_{1} ) \) being small enough. The results obtained are as follows and their proofs are not difficult with a procedure analogous to the ones in Theorems 2.4, 3.2.
Theorem 3.7
Suppose that \(2< p\leq\frac {2N-2}{N-2}\), \(K>0\), \(\lambda>0\), \(g\equiv0\), \(( u_{0},u_{1} ) \in H^{1}\times L^{2} \) and (\(\mathrm{A}_{1}^{\prime\prime}\)), (\(\mathrm{A}_{2}^{\prime \prime}\)) hold. Then problem (3.40) has a unique global solution \(u\in L^{\infty} ( 0,\infty;H^{1} ) \cap C ( [0,\infty);H^{1} ) \cap C^{1} ( [0,\infty);L^{2} ) \) such that \(u_{t}\in L^{\infty} ( 0,\infty;L^{2} ) \).
Furthermore, if \(\Vert h\Vert _{L^{\infty}(0,\infty ;L^{2} ( \partial\Omega\times\Omega ) )}\), \(\Vert h^{\prime} \Vert _{L^{\infty}(0,\infty;L^{2} ( \partial \Omega \times\Omega ) )}\) are sufficiently small then there exist positive constants C, γ such that
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The authors wish to express their sincere thanks to the referees for their valuable comments. The authors are also extremely grateful to Vietnam National University Ho Chi Minh City for the encouragement.
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Appendix
Appendix
Lemma A.1
Let H be Hilbert space with respect to the scalar product \(\langle\cdot,\cdot\rangle\). Then the embedding \(W^{1}(0,T;H)=\{F\in L^{2}(0,T;H):F^{\prime}\in L^{2}(0,T;H)\}\hookrightarrow C^{0}([0,T];H)\) is continuous and
for all \(F\in W^{1}(0,T;H)\), where \(\gamma_{T}=\sqrt{ \frac{1}{2T}+\sqrt{4+\frac{1}{4T^{2}}}}\).
Proof
Let \(F\in W^{1}(0,T;H)\), for all \(t, s\in[0,T]\), we have
Hence \(F\in C^{0}([0,T];H)\).
On the other hand
Integrating with respect to s, we obtain
Inverting the variables s and r in the last integral of (A.3), we rewrite it as follows:
By the inequality \(2ab\leq\alpha a^{2}+\frac{1}{\alpha}b^{2}\), for all \(a,b\in \mathbb{R}\), \(\alpha>0\), we deduce from (A.3), (A.4) that
Choose \(\alpha>0\) such that \(1+2T\alpha=\frac{2T}{\alpha}\), or \(\alpha=\frac{2}{\frac{1}{2T}+\sqrt{4+\frac{1}{4T^{2}}}}\). Hence
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Ngoc, L.T.P., Triet, N.A. & Long, N.T. Existence and exponential decay estimates for an N-dimensional nonlinear wave equation with a nonlocal boundary condition. Bound Value Probl 2016, 20 (2016). https://doi.org/10.1186/s13661-016-0527-5
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DOI: https://doi.org/10.1186/s13661-016-0527-5