Abstract
This work is concerned with a system of two singular viscoelastic equations with general source terms and nonlocal boundary conditions. We discuss the stabilization of this system under a very general assumption on the behavior of the relaxation function \(k_{i}\), namely,
We establish a new general decay result that improves most of the existing results in the literature related to this system. Our result allows for a wider class of relaxation functions, from which we can recover the exponential and polynomial rates when \(k_{i}(s) = s^{p}\) and p covers the full admissible range \([1, 2)\).
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1 Introduction
In this paper, we consider the following system:
where \(\Omega =(0,L)\), \(k_{i}:[0,+\infty )\longrightarrow (0,+\infty )\), (\(i=1,2\)), are non-increasing differentiable functions satisfying more general conditions to be mentioned later and
where \(r>-1\) and \(a,b>0\).
Mixed nonlocal problems for parabolic and hyperbolic partial differential equations have received a great attention during the last few decades. These problems are especially inspired by modern physics and technology. They aim to describe many physical and biological phenomena. For instance, physical phenomena are modeled by initial boundary value problems with nonlocal constraints such as integral boundary conditions, when the data cannot be measured directly on the boundary, but the average value of the solution on the domain is known. Initial boundary value problems for second-order evolution partial differential equations and systems having nonlocal boundary conditions can be encountered in many scientific domains and many engineering models and are widely applied in heat transmission theory, underground water flow, medical science, biological processes, thermoelasticity, chemical reaction diffusion, plasma physics, chemical engineering, heat conduction processes, population dynamics, and control theory. See in this regard the work by Cannon [1], Shi [2], Capasso and Kunisch [3], Cahlon and Shi [4], Ionkin and Moiseev [5], Shi and Shilor [6], Choi and Chan [7], and Ewing and Lin [8]. In early work, most of the research on nonlocal mixed problems was devoted to the classical solutions. Later, mixed problems with integral conditions for both parabolic and hyperbolic equations were studied by Pulkina [9, 10], Yurchuk [11], Kartynnik [12], Mesloub and Bouziani [13], Mesloub and Messaoudi [14, 15], Mesloub [16], and Kamynin [17]. For instance, Said Mesloub and Fatiha Mesloub [18] obtained existence and uniqueness of the solution to the following problem:
and proved that the solution blows up for large initial data and decays for sufficiently small initial data. Mesloub and Messaoud [14] considered the following nonlocal singular problem:
and proved blow-up result for large initial data and decay results of sufficiently small initial data enough for \(p>2\). In [19], Draifia et al. proved a general decay result for the following singular one-dimensional viscoelastic system:
where \(Q=(0,\alpha )\times (0,t)\) and \(p,q >1\). Piskin and Ekinci [20] studied problem (1) when the Bessel operator has been replaced by a Kirchhoff operator with a degenerate damping terms. They proved the global existence and established a decay rate of solution and also a finite time blow up. Recently, Boulaaras et al. [21] treated problem (1) and proved the existence of a global solution to the problem using the potential-well theory. Moreover, they established a general decay result in which the relaxation functions \(k_{1}\) and \(k_{2}\) satisfy
Motivated by the above work, we prove a general stability result of system (1) replacing the condition (6) used in [21] by a more general assumption of the form:
Our decay result improves all the existing results in the literature related to this system.
This paper is divided into four sections. In Sect. 2, we state some assumptions needed in our work. Some technical lemmas will be given in Sect. 3. The statement with proof of the main result and some examples will be given in Sect. 4.
2 Preliminaries
In this section, we present some materials needed in the proof of our results. We also state, without proof, the global existence result for problem (1). Let \(L^{p}_{x}=L^{p}_{x}(0,L)\) be the weighted Banach space equipped with the norm
\(L^{2}_{x}(0,L)\) is the Hilbert space of square integral functions having the finite norm
\(V=V^{1}_{x}(0,L)\) is the Hilbert space equipped with the norm
and
Lemma 2.1
([14])
\(\forall w\in V_{0}\), a Poincaré-type inequality is
Remark 2.1
Notice that \(\Vert u \Vert _{V_{0}}=\Vert u_{x} \Vert _{L^{2}_{x}}\) defines an equivalent norm on \(V_{0}\).
2.1 Assumptions
- \((A1)\) :
-
\(k_{i}:\mathbb{R}_{+}\to \mathbb{R}_{+}\) (for \(i=1,2\)) are \(C^{1}\) non-increasing functions satisfying
$$\begin{aligned} k_{i}(0)>0,\quad 1- \int _{0}^{+\infty }k_{i}(s)\,ds=:\ell _{i}>0. \end{aligned}$$(8) - \((A2)\) :
-
There exist non-increasing differentiable functions \(\xi _{i}:[0,+\infty )\longrightarrow (0,+\infty )\) and \(\boldsymbol{C}^{1}\) functions \(\Psi _{i}:[0,+\infty )\longrightarrow [0,+\infty )\) which are linear or strictly increasing and strictly convex \(\boldsymbol{C}^{2}\) functions on \((0,\varepsilon ]\), \(\varepsilon \leq k_{i}(0)\), with \(\Psi _{i}(0)=\Psi _{i}'(0)=0\) such that
$$\begin{aligned} k_{i}'(t)\leq -\xi _{i}(t)\Psi _{i}\bigl(k_{i}(t)\bigr),\quad \forall t\geq 0 \text{ and for }i=1,2. \end{aligned}$$(9)
Remark 2.2
The given functions \(f_{1}\) and \(f_{2}\) satisfy
where
Lemma 2.2
(Jensen’s inequality)
Let \(G:[a,b]\longrightarrow \mathbb{R}\) be a convex function. Assume that the functions \(f:(0,L)\longrightarrow [a,b]\) and \(h:(0,L)\longrightarrow \mathbb{R}\) are integrable such that \(h(x)\geq 0\), for any \(x\in (0,L)\) and \(\int _{0}^{L}h(x)\,dx=k>0\). Then
Remark 2.3
If Ψ is a strictly increasing, strictly convex \(C^{2}\) function over \((0, \varepsilon ]\) and satisfying \(\Psi (0) = \Psi '(0) = 0\), then it has an extension, Ψ̅, that is also strictly increasing and strictly convex \(C^{2}\) over \((0,\infty )\). For example, if \(\Psi (\varepsilon ) = a, \Psi '(\varepsilon ) = b, \Psi ''( \varepsilon ) = c\), and for \(t > \varepsilon \), Ψ̅ can be defined by
Remark 2.4
Since \(\Psi _{i}\) is strictly convex on \((0,\varepsilon ]\) and \(\Psi _{i}(0)=0\),
The modified energy functional E associated to problem (1) is
where, for any \(w\in L^{2}_{loc} ([0,+\infty );L_{x}^{2}(0,L) )\) and \(i=1,2\),
Using (1) with direct differentiation gives
2.2 Local and global existence
In this subsection, we state, without proof, the local and global existence results for system (1), which can be proved similarly to the ones in [14, 18] and [21].
Theorem 2.1
Assume that \((A1)\) and \((A2)\) hold. If \((u_{0},v_{0})\in V_{0}^{2}\) and \((u_{1},v_{1}) \in (L_{x}^{2})^{2} \). Then problem (1) has a unique local solution.
For the global existence, we introduce the following functionals:
and
We notice that \(E(t)=J(t)+\frac{1}{2}\Vert u_{t} \Vert ^{2}_{L^{2}_{x}}+ \frac{1}{2}\Vert v_{t} \Vert ^{2}_{L^{2}_{x}}\).
Lemma 2.3
Suppose that \((A1)\) and \((A2)\) hold. Then, for any \((u_{0},v_{0})\in V_{0}^{2}\) and \((u_{1},v_{1}) \in (L^{2}_{x})^{2} \) satisfying
there exists \(t_{*} > 0\) such that
Remark 2.5
We can easily deduce from Lemma 2.3 that
Theorem 2.2
Assume that \((A1)\) and \((A2)\) hold. If \((u_{0},v_{0})\in V_{0}^{2}\) and \((u_{1},v_{1}) \in (L_{x}^{2})^{2} \) and satisfies (16), then the solution of (1) is global and bounded.
3 Technical lemmas
In this section, we establish several lemmas needed for the proof of our main result.
Lemma 3.1
There exist two positive constants \(c_{1}\) and \(c_{2}\) such that
Proof
We prove inequality (19) for \(f_{1}\) and the same result holds for \(f_{2}\). It is clear that
From (20) and Young’s inequality, with
we get
hence
Consequently, by using (7), (12), (13) and the embedding \(V_{0} \hookrightarrow L^{2(2r+3)}\), we obtain
This completes the proof of Lemma 3.1. □
Lemma 3.2
([22])
There exist positive constants d and \(t_{0}\) such that, for any \(t\in [0,t_{0}]\), we have
Lemma 3.3
If \((A1)\) holds. Then, for any \(w\in V_{0}\), \(0<\alpha <1\) and \(i=1,2\), we have
where \(C_{\alpha,i}:=\int _{0}^{\infty }\frac{k_{i}^{2}(s)}{\alpha k_{i}(s)-k_{i}'(s)}\,ds\) and \(h_{i}(t):=\alpha k_{i}(t)-k_{i}'(t)\).
Proof
The proof of this lemma goes similar to the one in [22]. □
Lemma 3.4
Under the assumptions \((A1)\) and \((A2)\), the functional
satisfies, along with the solution of system (1), the estimate
Proof
Direct differentiation, using (1), yields
Using Young’s inequality, we obtain, for any \(\delta _{1}, \delta _{2}\in (0,1)\),
Taking \(\delta _{1}=\ell _{1}\) and \(\delta _{2}=\ell _{2}\) and using Lemma 3.3, we have
□
Let us introduce the functionals
and
Lemma 3.5
Assume that \((A1)\) and \((A2)\) hold. Then the functional
satisfies, along with the solution of (1), the following estimate:
where \(0<\delta <1\).
Proof
Direct differentiation, using (1), gives
Using Young’s inequality and Lemma 3.3, we get, for any \(0<\delta <1\), the following:
Using Young’s inequality, (18), (19) and (22), we have
Also, by applying Young’s inequality and Lemma 3.3, we obtain, for any \(0<\delta <1\),
Similarly, we have
A combination of all the above estimates gives
Repeating the same calculations with \(\chi _{2}\), we obtain
Therefore, (33) and (34) imply (27), which completes the proof of Lemma 3.5. □
Lemma 3.6
Assume that \((A1)\) and \((A2)\) hold. Then the functionals \(J_{1}\) and \(J_{2}\) defined by
and
satisfy, along with the solution of (1), the estimates
where \(K_{i}(t):=\int _{t}^{\infty }k_{i}(s)\,ds\) (for \(i=1,2\)) and \(\ell =\min \{\ell _{1},\ell _{2}\}\).
Proof
We will prove inequality (35) and the same proof also holds for (36). By Young’s inequality and the fact that \(K_{1}^{\prime }(t)=-k_{1}(t)\), we see that
Now,
Using the facts that \(K_{1}(0)=1-\ell _{1}\) and \(\int _{0}^{t}k_{1}(s)\,ds \le 1-\ell _{1}\), (35) is established. □
Lemma 3.7
The functional L defined by
satisfies, for a suitable choice of \(N,N_{1},N_{2}\ge 1\),
and the estimate
where \(t_{0}\) is introduced in Lemma 3.2and \(\ell =\min \{\ell _{1},\ell _{2}\}\).
Proof
It is not difficult to prove that \(L(t)\sim E(t)\). To establish (38), we choose \(\delta =\frac{\ell }{4cN_{2}}\) where \(\ell =\min \{\ell _{1},\ell _{2}\}\). We set \(C_{\alpha }=\max \{C_{\alpha,1},C_{\alpha,2}\}\) and \(k_{0}=\min \lbrace \int _{0}^{t_{0}}k_{1}(s)\,ds,\int _{0}^{t_{0}}k_{2}(s)\,ds \rbrace >0\). Now using (23) and (28) and recalling the fact that \(k_{i}'=\alpha k_{i}-h_{i}\), we obtain, for any \(t\geq t_{0}\),
First, we choose \(N_{1}\) so large such that \(\frac{\ell }{4}(2N_{1}-1)>4(1-\ell )\).
Then we select \(N_{2}\) large enough so that \(k_{0} N_{2}-\frac{\ell }{4c}-N_{1}>1\). Now, one can use the Lebesgue dominated convergence theorem with the fact that \(\frac{\alpha k_{i}^{2}(s)}{\alpha k_{i}(s)-k_{i}'(s)}< k_{i}(s)\), for \(i=1,2\), to prove that
Therefore, there exists \(\alpha _{0}\in (0,1)\) such that if \(\alpha <\alpha _{0}\), then, we get \(\alpha C_{\alpha }< \frac{1}{8 [\frac{4c^{2}}{\ell }N^{2}_{2}+cN_{1} ]}\). Then, by letting \(\alpha =\frac{1}{2N}<\alpha _{0}\), we get \(\frac{1}{4}N-\frac{4c^{2}}{\ell }N^{2}_{2}>0\). This leads to
Then, (38) is established. □
4 General decay result
In this section, we state and prove our main result.
Theorem 4.1
Let \((u_{0},v_{0}) \in V_{0}^{2}\) and \((u_{1},v_{1})\in (L_{x}^{2})^{2}\) be given and satisfying (16). Assume that \((A1)\) and \((A2)\) hold. If \(\Psi _{1}\) and \(\Psi _{2}\) are linear, then there exist two positive constants \(\lambda _{1}\) and \(\lambda _{2}\) such that the solution to problem (1) satisfies the estimate
where \(t_{0}\) is introduced in Lemma 3.2and \(\xi (t)=\min \{\xi _{1}(t),\xi _{2}(t)\}\).
Proof
Using (21) and (13) we have, for any \(t\geq t_{0}\),
Using this inequality, the estimate (38) becomes, for some \(m>0\) and for any \(t\geq t_{0}\),
Let \(\mathcal{L}:= L+cE\sim E\), we obtain
Multiply both sides of (40) by \(\xi (t)=\min \{\xi _{1}(t),\xi _{2}(t)\}\) where ξ is non-increasing function and using \((A2)\) and (13) we get, for any \(t\geq t_{0}\) and \(m>0\), the following:
Since ξ is non-increasing, we have
Integrating over \((t_{0},t)\) and using the fact that \(\xi \mathcal{L}+cE\sim E\), then, for any \(\lambda _{1},\lambda _{2} > 0\), we obtain
□
Theorem 4.2
Let \((u_{0},v_{0}) \in V_{0}^{2}\) and \((u_{1},v_{1})\in (L_{x}^{2})^{2}\) be given and satisfying (16). Assume that \((A1)\) and \((A2)\) hold. If \(\Psi _{1}\) or \(\Psi _{2}\) is nonlinear, then there exist two positive constants \(\lambda _{1}\) and \(\lambda _{2}\) such that the solution to problem (1) satisfies the estimate
where
Proof
Using Lemmas 3.6 and 3.7, we easily see that
is nonnegative and, for any \(t\geq t_{0}\), and, for some \(C>0\),
Therefore, we arrive at
Now, we define the following functionals:
Thanks to (42), one can choose \(0<\gamma <1\) so that
Without loss of the generality, we assume that \(I_{i}(t)>0\), for any \(t> t_{0}\); otherwise, we get an exponential decay from (38). We also define the following functionals:
and observe that
Using (2.4), Assumption \((A2)\), inequality (43) and Jensen’s inequality, we obtain
where \(\bar{\Psi }_{1}\) is defined in Remark (2.3). Then, we have
Similarly, we can have
Thus, the estimate (40) becomes
Set \(H=\min \{\bar{\Psi }_{1}',\bar{\Psi }_{2}'\}\) and define the functional
Using the fact that \(\bar{\Psi }_{i}'>0\), \(\bar{\Psi }_{i}''>0\) and \(E'\leq 0\), we also deduce that \(F_{1}\sim E\). Further, we get
Recalling that \(E'\leq 0\), then we drop the first and last terms of the above identity. Therefore, by using the estimate (45), we have
In the sense of Young [23], we let \(\bar{\Psi }_{i}^{*}\) be the convex conjugate of \(\bar{\Psi }_{i}\) such that
and it satisfies the following generalized Young inequality:
By letting \(A=H (\varepsilon _{0}\frac{E(t)}{E(0)} )\), \(B_{i}=\bar{\Psi }_{i}^{-1} ( \frac{\gamma \eta _{i}(t)}{\xi _{i}(t)} )\), for \(i=1,2\), and combining (46)–(48), we have, for almost every \(t\geq t_{0}\),
Multiplying the above estimate by \(\xi (t)=\min \{\xi _{1}(t),\xi _{2}(t)\}>0\) and using the fact in (44), we get
Select \(\varepsilon _{0}\) small enough so that \(k_{0}:= mE(0)-c\varepsilon _{0} >0\), and we obtain
Let \(F_{2}=\xi F_{1}+cE\sim E\), we have, for some \(\alpha _{1},\alpha _{2}>0\), the following equivalent inequality:
Hence, we have
Now, we set
Using the fact that \(\Psi _{i}'>0\) and \(\Psi _{i}''>0\) on \((0,r]\) (for \(i=1,2\)), we deduce that \(H_{0},H_{0}'>0\) \(a.e\). on \((0,1]\). Now, we define the following functional:
and use (49) and (50) to show that \(R\sim E\) and, for some \(\beta _{1}>0\),
Integrating over the interval \((t_{0},t)\) and using a change of variables, we get
which gives
where \(\Psi _{*}(t):=\int _{t}^{r}\frac{1}{s H(s)}\,ds\). Since \(R\sim E\), we have, for \(\beta _{2}>0\),
This completes the proof. □
Example 4.3
-
(1)
Let \(k_{1}(t)=ae^{-\alpha t}\) and \(k_{2}(t)=\frac{b}{(1+t)^{q}}\), \(q>1\). The constants a and b are chosen so that \((A1)\) is satisfied. Then there exists \(C>0\) such that
$$\begin{aligned} E(t)\leq \frac{C}{(1+t)^{q}},\quad \forall t>0. \end{aligned}$$ -
(2)
Let \(k_{1}(t)=\frac{a}{(1+t)^{m}}\) and \(k_{2}(t)=\frac{b}{(1+t)^{n}}\) with \(m,n>1\). The constants a and b are chosen so that \((A1)\) is satisfied. Then there exists \(C>0\) such that, for any \(t>0\),
$$\begin{aligned} E(t)\leq \frac{C}{(1+t)^{\nu }}, \quad\text{with }\nu =\min \{m,n \}. \end{aligned}$$ -
(3)
Let \(k_{1}(t)=ae^{-\beta t}\) and \(k_{2}(t)=be^{-(1+t)^{q}}\) with \(0< q<1\). The constants a and b are chosen so that \((A1)\) is satisfied. Then there exist positive constants C and \(\alpha _{1}\) such that
$$\begin{aligned} E(t)\leq Ce^{-\alpha _{1}(1+t)^{\nu }}, \quad\text{for } t \text{ large}. \end{aligned}$$
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Acknowledgements
The authors would like to express their profound gratitude to King Fahd University of Petroleum and Minerals (KFUPM) and University of Sharjah for their continuous support and he also thanks an anonymous referee for his/her very careful reading and valuable suggestions. This work is funded by KFUPM under Project #SB191048.
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Al-Gharabli, M.M., Al-Mahdi, A.M. & Messaoudi, S.A. New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions. Bound Value Probl 2020, 170 (2020). https://doi.org/10.1186/s13661-020-01467-5
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DOI: https://doi.org/10.1186/s13661-020-01467-5