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Global existence and exponential decay of solutions for generalized coupled non-degenerate Kirchhoff system with a time varying delay term
Boundary Value Problems volume 2020, Article number: 90 (2020)
Abstract
The paper studies a system of nonlinear viscoelastic Kirchhoff system with a time varying delay and general coupling terms. We prove the global existence of solutions in a bounded domain using the energy and Faedo–Galerkin methods with respect to the condition on the parameters in the coupling terms together with the weight condition as regards the delay terms in the feedback and the delay speed. Furthermore, we construct some convex function properties, and we prove the uniform stability estimate.
1 Introduction
The Kirchhoff equation belongs to the famous wave equation’s models describing the transverse vibration of a string fixed in its ends. It has been introduced in 1876 by Kirchhoff [8] and it is more general than the D’Alembert equation. In one dimensional space it takes the following form:
where the function \(u(x,t)\) is the vertical displacement at the space coordinate x, varying in the segment \([0,L]\) and over time \(t>0\), ρ is the mass density, h is the area of the cross section of the string, \(P_{0}\) is the initial tension on the string, L is the length of the string and E is the Young modulus of the material. The nonlinear coefficient
is obtained by the variation of the tension during the deformation of the string. When we do not have an initial tension (i.e. \(P_{0}=0\)), we call that a degenerate case as opposed to the non-degenerate case.
In this paper, we are interested in studying, in \(\mathcal{A}=\varOmega \times (0,\infty )\), the following coupled viscoelastic Kirchhoff system:
in which Ω is an n dimensional bounded domain of \(\mathbb{R}^{n}\) and we have a smooth boundary Γ, \(l>0\), \(\mu _{1}\) and \(\mu _{2}\) are positive real constants, \(h_{1}\) and \(h_{2}\) are positive functions with exponential decay, and \(\tau (t)\) is a positive time varying delay. In addition the initial condition \((u_{0},v_{0},u_{1},v_{1},f_{0},g_{0})\) will be specified in their function space later. M is a smooth function defined by
with \(a,b>0\), and \(\gamma \geq 1\). \(f_{1}\) and \(f_{2}\) are two functions taking a particular form that we will make precise later.
The problem (1.2) is a description of axially moving viscoelastic strings composed of two different materials (like the wires of electricity) that are nonhomogeneous and which will be of influence on its moving, specially on the acceleration. From the mathematical point of view, this influence is represented by \(\vert w_{t} \vert ^{l}w^{{\prime \prime }}\), where \(\vert w_{t} \vert ^{l}\) is the material density, varying the velocity. A lot of work has been published with this term, for example see [11] and [14], where we find different results about the global existence and nonexistence of solutions and the decay of energy.
In recent years, the study of wave equations with delay has become an active area and with different forms of delay (constant [7], switching [5], varying in time [12], distributed [6]). The delay appears in modeling of a lot of domains, like the physical, chemical, biological and engineering domains. It is introduced when we have a time lag between an action on a system and a response of the system to this action. Furthermore, a delay can be small enough in feedback yet can destabilize a system [10], or improve the performance of the system [17].
In the absence of delay, Cavalcanti et al. [3] studied the following viscoelastic wave equations with strong damping:
They used the Fadeo–Galerkin method to prove the global existence of a solution; also an explicit decay rate of the energy has been given provided \(m>0\).
In the other hand, in the same case and for \(l=0\), Raslan et al. [16] and El-Sayed et al. [4] have studied coupled equal width wave equations with strong damping, as they were looking for the new exact solution.
The problem treated in [2] has the following form:
Under the assumptions set on \(g_{1}\), \(g_{2}\), σ and τ, the authors have gotten the global existence of a solution and the decay rate of the energy.
Recently, Mezouar and Boulaaras [13] have studied the viscoelastic non-degenerate Kirchhoff equation with varying delay term in the internal feedback.
In the present paper, we extend our recently published paper in [13] for a coupled system (1.2). The famous technique of using the presence of a delay in the PDE problem is to set a new variable defined by a velocity dependent on the delay, which will give us a new problem equivalent to our studied problem; but the last one is a coupled system without delay. After this, we can prove the existence of global solutions in suitable Sobolev spaces by combining the energy method with the Fadeo–Galerkin procedure and under the choice of a suitable Lyapunov functional, we establish an exponential decay result.
The outline of the paper is as follows: In the second section, some hypotheses related to the problem are given and we state our main result. Then in the third section, the global existence of weak solutions is proven. Finally, in the fourth section, we give the uniform energy decay.
1.1 Preliminaries and assumptions
Similar to that [12], we present the new variables
and
Then we have
In the same way, we have
Therefore, problem (1.2) is equivalent to
Throughout this work and for simplifying our formulas, we will adopt the notation \(z_{i}\), u and v instead of \(z_{i}(x,\rho ,t)\), \(u(x,t)\) and \(v(x,t)\), except if that makes things inconvenient.
In order to demonstrate the main result in this paper, a few assumptions are needed.
- (A-1):
Consider that \(0< l\leq \gamma \) verifies
$$ \textstyle\begin{cases} \gamma \leq \frac{2}{n-2}&\text{in the case } n > 2, \\ \gamma < \infty & \text{in the case } n\leq 2. \end{cases} $$- (A-2):
As regards the relaxation functions \(h_{i}:{\mathbb{R}_{+}}\rightarrow {\mathbb{R}_{+}}\) we see that they are bounded \(C^{1}\) functions such that
$$ a- \int _{0}^{\infty }h_{i}(s)\,ds\geq k>0. $$We assume also that there exist some positive constants \(\zeta _{i} \) verifying
$$ h_{i}^{\prime }(t)\leq -\zeta _{i} h_{i}(t) $$for \(i=1,2\).
- (A-3):
We have \(\tau \in C^{2}([0,T],[\tau _{0},\tau _{1}])\) a positive function, where
$$ \tau ^{\prime }(t)\leq d< 1,\quad \forall t\in [0,T]. $$- (A-4):
\(f_{1}(u,v)=\alpha v+b_{1} \vert v \vert ^{q+1} \vert u \vert ^{p-1}u\) and \(f_{2}(u,v)=\alpha u+b_{2} \vert u \vert ^{p+1} \vert v \vert ^{q-1}v\) where \(\alpha >0\), \(b_{1}=(p+1)(p+q)\), \(b_{2}=(q+1)(p+q)\) such that p and q are conjugate (i.e. \(\frac{1}{p}+\frac{1}{q}=1\)), \(p,q<\gamma -\frac{1}{2}\) and satisfy
$$ 2\leq p,q\leq \textstyle\begin{cases} \sqrt{\frac{n}{2(n-2)}}& \text{if } n> 2, \\ +\infty &\text{if } n\leq 2. \end{cases} $$
The energy related to the system solution of (1.5) is defined as follows:
where ξ is a positive constant such that
and
Lemma 1.1
(Sobolev–Poincaré’s inequality)
Let q be a number with
Then there exists a constant \(C_{s}=C_{s}(\varOmega ,q) \) such that
We present the following lemma.
Lemma 1.2
[15] Forh, φ\(C^{1}\)-real functions, we have
Lemma 1.3
Let\((u,v,z_{1},z_{2})\)be a solution of the problem (1.5). Then the energy functional defined by (1.6) satisfies
where\(\lambda =\xi +\frac{\mu }{2} \), \(\beta =\xi (1-d)-\frac{\mu }{2}\)and\(\mu =\max \{\mu _{1},\mu _{2}\}\)are positive.
Proof
After the multiplication of the first equation in (1.5) by \(u_{t}\) followed by integration of the result by parts over Ω, we get
Using (1.8) and (1.10) leads to
Similarly by multiplying the second equation in (1.5) by \(v_{t}\), integrating over Ω and using integration by parts, we get
Multiplying the third equation in (1.5) by \(\xi \Delta z_{1}\) and integrating the result over \(\varOmega \times (0,1)\), we obtain
Consequently,
Similarly we get
Combining (1.11)–(1.14), taking the derivation of energy leads to
From \((A3)\), we find the following bound:
Using (1.7), we complete the proof of the lemma. □
2 Global existence
Theorem 2.1
Let\((u_{0},v_{0})\in (H^{2}(\varOmega ) \cap H_{0}^{1}(\varOmega ))^{2}\), \((u_{1},v_{1})\in (H_{0}^{1}(\varOmega ))^{2}\)and\((f_{0},g_{0})\in (H_{0}^{1}(\varOmega , H^{1}(0,1)))^{2}\)satisfy the compatibility condition
Assume that(A1)–(A3)hold. Then the problem (1.2) admits a weak solution such that\(u,v\in L^{\infty }(0,\infty ;H^{2}(\varOmega ) \cap H_{0}^{1}(\varOmega ))\), \(u_{t},v_{t}\in L^{\infty }(0,\infty ; H_{0}^{1}(\varOmega ))\), and\(u_{tt},v_{tt}\in L^{2}(0,\infty ,H_{0}^{1}(\varOmega ))\).
Proof
As in the previous assumptions in [2] for the initial conditions \(u_{0},v_{0}\in H^{2}(\varOmega )\cap H_{0}^{1}(\varOmega )\), \(u_{1},v_{1}\in H_{0}^{1}( \varOmega )\), \(f_{0},g_{0}\in H_{0}^{1}(\varOmega ,H^{1}(0,1))\) and the basic functions, we introduce the approximate solutions \((u^{k},v^{k},z_{1}^{k},z_{2}^{k})\), \(k=1,2,3,\ldots \) , in the form
where \(a^{jk}\), \(b^{jk}\), \(c^{jk}\) and \(d^{jk}\) (\(j=1,2,\ldots,k\)) are determined by the following ordinary differential equations:
Also
Noting that \(\frac{l}{2(l+1)}+\frac{1}{2(l+1)}+\frac{1}{2}=1\), by applying the generalized Hölder inequality, we find
Since \((A1)\) holds, according to the Sobolev embedding the nonlinear terms \(( \vert u^{k}_{t} \vert ^{l}u^{k}_{tt},w_{j})\) and \(( \vert v^{k}_{t} \vert ^{l}v^{k}_{tt},w_{j})\) in (2.1) make sense (see [2]).
A. First estimate.
Since the sequences \(u^{k}_{0}\), \(v^{k}_{0}\), \(u^{k}_{1}\), \(v^{k}_{1}\), \(z^{k}_{1}(\rho ,0)\) and \(z^{k}_{2}(\rho ,0)\) converge and from Lemma 1.3 with employing Gronwall’s lemma, we find \(C_{1}>0\) independent of k such that
where
Noting \((A1)\) and the estimate (2.9) yields
B. The second estimate.
By multiplying the first side of equation (respectively, the second equation) in (2.1) by \(a_{tt}^{jk}\) (respectively, by \(b_{tt}^{jk}\)), by summing j from 1 to k, then
Differentiating (2.6) with respect to t, we get
Multiplying the first equation by \(c_{t}^{jk}\) (respectively the second equation by \(d_{t}^{jk}\)), summing over j from 1 to k, we have
Integrating over \((0,1)\) with respect to ρ, we obtain
Summing (2.13), (2.14) and as \(M(r)\geq a\), we get
We estimate the right hand side of (2.15) as follows:
From the integration by parts, we have
Using the inequality \(ab\leq \frac{1}{2}a^{2}+\frac{1}{2}b^{2}\) and Sobolev–Poincaré inequalities, we obtain
On the other hand, by recalling (A-4) and Lemma 1.1 and using Young’s inequality, we get
Hence from summing (2.16) and (2.17) we deduce that
Similarly
Also by Young’s inequality, we get
We have
Similarly
and
Taking into account (2.18)–(2.23) into (2.15) yields
By using (A3) and taking the first estimate (2.9) into account, we infer
where \(C_{2}\) is a positive constant that depends on η, α, a, \(C_{s}\), \(\vert \varOmega \vert \), \(b_{1}\), \(b_{2}\), p, q, \(C_{1}\) for \(i=1,2\).
Integrating (2.24) over (0,t) we obtain
For a suitable \(\eta >0\) such that \(1-(\eta (\mu _{i}^{2}+2)+\frac{(1+b_{i})C_{s}^{2}}{2})>0\) for \(i=1,2\), we obtain the second estimate
We observe from the estimate (2.9) and (2.25) that there exist subsequences \((u^{m})\) of \((u^{k})\) and \((v^{m})\) of \((v^{k})\) such that
In the following, we will treat the nonlinear term. From the first estimate (2.9) and Lemma 1.1, we deduce
where \(C_{4}\) depends only on \(C_{s}\), \(C_{1}\), T, l.
On the other hand, from the Aubin–Lions theorem (see Lions [9]), we deduce that there exists a subsequence of \((u^{m})\), still denoted by \((u^{m})\), such that
which implies
Hence
where \(\mathcal{A}=\varOmega \times (0,T)\). Thus, using (2.31), (2.33) and the Lions lemma, we derive
similarly
and
which implies \((z_{1}^{m},z_{2}^{m})\rightarrow (z_{1},z_{2})\) almost everywhere in \(\mathcal{A}\).
The sequences \((u^{m})\) and \((v^{m})\) satisfy
and
we have
As we add and subtract \(\vert v^{k} \vert ^{q+1} \vert u \vert ^{p}u\) to the previous formula, we obtain
We use the following elementary inequalities:
and
for some constant C, \(\forall k\geq 1\) and \(\forall a,b\in \mathbb{R}\). Hence (2.38) becomes
The typical term in the above formula can be estimated as follows.
Noting that \(\frac{l}{2p}+\frac{1}{2q}+\frac{1}{2}=1\), by applying the generalized Hölder inequality, we find
Recalling \((A4)\), Lemma 1.1 and (2.9), we get
Hence (2.39) yields
As \((u^{m})\), \((v^{m})\) are Cauchy sequences in \(L^{\infty }(0,T,H_{0}^{1}(\varOmega ))\) (we prove it as in [1]) then we deduce (2.36). Similarly we get the convergence (2.37).
By multiplying (2.1) and (2.6) by \(\theta (t)\in \mathcal{D}(0,T)\) and by integrating over \((0,T)\), it follows that
for all \(j=1,\ldots,k\).
The convergence of (2.26)–(2.30), (2.35), (2.34), (2.36) and (2.37) is sufficient to pass to the limit in (2.43). This completes the proof of the theorem. □
3 Exponential decay rate
In order to make precise the asymptotic behavior of our solutions, we introduce some functionality to determine a suitable Lyapunov functional equivalent to E.
Theorem 3.1
Assume that(A1)–(A3)hold. Then for every\(t_{0}>0\)there exist positive constantsKand\(c^{\prime }\)such that the energy defined by (1.6) obeys the following decay:
Lemma 3.2
Along a solution of the problem (1.5) the functional
satisfies the following estimates:
Proof
(ii) A direct derivation of (3.2) gives
Because the exponential function \(e^{-2\rho \tau (t)}\) decreases on \((0,1)\times (\tau _{0},\tau _{1}) \) and from \((A3)\), we get the results of this lemma. □
Lemma 3.3
Along a solution of the problem (1.5) the functional
verifies the estimates
and
Proof
(i) Applying Young’s inequality, Sobolev–Poincaré’s inequality and \(L^{l+2}\hookrightarrow L^{2}\), we find
(ii) Taking a direct derivation of (3.2) and replacing \(\vert u_{t} \vert ^{l}u_{tt}\), \(\vert v_{t} \vert ^{l}v_{tt}\) from the first and seconde equations of (1.5) give
As \(M(r)\geq a\) and making use of Young’s inequality we obtain
By use of Young’s inequality, the third term in the right side is estimated as follows:
Similarly
and from \((A4)\)
Thus, (3.6) is valid. □
Lemma 3.4
Along a solution of the problem (1.5) the functional
satisfies the estimates
and
where\(\delta >0\)and\(c_{s}\)is the Sobolev embedding constant.
Proof
We have
We use Young’s inequality with the conjugate exponents \(p^{\prime }=\frac{l+2}{l+1}\) and \(q^{\prime }=l+2\), then the second term in the right hand side can be estimated as
We get by using Hölder’s inequality
Combining (3.12) with (3.11) we obtain
In the same way, we get
Similarly
Combining (3.13),(3.14) and (3.15), we deduce (i).
(ii) We use the Leibnitz formula and the first and second equations of (1.5) to find
where
and
Next we will estimate \(I_{1},\ldots,I_{6}\).
For \(I_{1}\), by applying Hölder’s and Young’s inequalities, we obtain
Similarly,
and using the fact that \(l\leq \gamma \)
For \(I_{6}\), we have
and
By using the Young and Hölder inequalities and Lemma 1.1, we find
Also by following a similar technique to above, we get
Hence
Similarly
Summing (3.22), (3.26) and (3.27), we get
Combining (3.16) and (3.17)–(3.28), we complete the proof. □
Proof of Theorem 3.1
Now, for \(M,\varepsilon _{1}>0\), we introduce the following functional:
Firstly we prove that \(F(t)\) is equivalent to \(E(t)\); for this we show that \(F(t)\) verified the following boundedness:
for some positive constants \(\kappa _{2}\), \(\kappa _{2}\).
We recall (3.3), (3.5), and (3.9) and, using the fact that \(l\leq \gamma \), we get
where \(\kappa >0\) depending the \(\varepsilon _{1}\), a, b, l, c, \(c_{s}\), k, ξ. For the choice of \(M=\kappa +\epsilon \) with \(\epsilon >0\), we get \(F(t)\sim E(t) \).
By recalling (1.9), (3.4), (3.6), (3.10) and (A2), we deduce that
where \(M_{0}=\max \{M \Vert \nabla u \Vert ^{2},M \Vert \nabla v \Vert ^{2}\}\), \(h_{0}=\min \{\int _{0}^{t_{0}}h_{1}(s)\,ds,\int _{0}^{t_{0}}h_{2}(s)\,ds\}\), \(h_{1}=\min \{h_{1}(0), h_{2}(0)\}\), \(h_{2}=\max \{h_{1}(0),h_{2}(0)\}\), \(\omega =\max \{b_{1}\frac{c_{s}^{4(q+1)}}{2}+\frac{c_{s}^{4q}}{2}b_{2},b_{2}\frac{c_{s}^{4(p+1)}}{2}+\frac{c_{s}^{4p}}{2}b_{1}\}\) and \(\zeta =\max \{\zeta _{1}, \zeta _{1}\}\).
Let \(\epsilon >0\) be sufficiently small so M is fixed, we take \(h_{0}-M\lambda -1>\varepsilon _{1}\) and δ small enough such that
Further, we choose η small enough such that
and
Thus
where \(m=\min \{\frac{2e^{-2\tau _{1}}}{\xi },2\frac{a_{2}}{a},a_{3}\}\) and \(c=\min \{\frac{a_{1}}{\beta },\frac{a_{4}}{\lambda },-2a_{5}\}\).
Let \(L(t)=F(t)+cE(t)\sim E(t)\). From (3.31), we get
for some \(c^{\prime }>0\). A simple integration over \((t_{0},t)\) yields
Thanks to the equivalence between L and E, we obtain (3.1). □
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Mezouar, N., Boulaaras, S. Global existence and exponential decay of solutions for generalized coupled non-degenerate Kirchhoff system with a time varying delay term. Bound Value Probl 2020, 90 (2020). https://doi.org/10.1186/s13661-020-01390-9
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DOI: https://doi.org/10.1186/s13661-020-01390-9