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Blow up, growth, and decay of solutions for a class of coupled nonlinear viscoelastic Kirchhoff equations with distributed delay and variable exponents
Journal of Inequalities and Applications volume 2024, Article number: 55 (2024)
Abstract
In this work, we consider a quasilinear system of viscoelastic equations with dispersion, source, distributed delay, and variable exponents. Under a suitable hypothesis the blow-up and growth of solutions are proved, and by using an integral inequality due to Komornik the general decay result is obtained in the case of absence of the source term \(f_{1}=f_{2}=0\).
1 Introduction
Our understanding of real-world phenomena and our technology today are largely based on mathematical analysis for partial differential equations (PDEs) [1, 2, 4, 5]. This mathematical analysis helps us to visualize and understand different real-world problems [7, 8, 10, 11]. The mathematical analysis study of PDEs has also taught us to show a little modesty: we have discovered the impossibility of predicting certain phenomena governed by nonlinear PDEs in the medium term—think of the now famous butterfly effect: a small variation of the initial conditions can lead to very large variations in very long time. On the other hand, we have also learned to “hear the shape of a drum”: it has been shown mathematically that the frequencies emitted by a drum during membrane vibration—a phenomenon described by a PDE—allow the drum shape to be perfectly reconstructed. One of the things to keep in mind about PDEs is that you usually do not want to get their solutions explicitly! What mathematics can do, on the other hand, is to say whether one or more solutions exist, and sometimes to very precisely describe certain properties of these solutions. However, the emergence of extremely powerful computers today makes it possible to obtain approximate solutions for partial derivative equations, even very complicated. This is what happens, for example, when you look at the weather forecast, or when we see the moving images of a simulation of airflow on the wing of airplane. The role of mathematicians is then to build approximation schemes and to demonstrate the relevance of the simulations by establishing a priori estimates on the made errors. When did EDP appear? They likely originated in the early days of rational mechanics in the seventeenth century, with figures like Newton and Leibniz playing crucial roles. As scientific disciplines, especially physics, advanced in energy functional, fluid mechanics equations, Navier–Stokes equations, where they contributed to the expansion of partial differential equations (PDEs).
To highlight a few key contributors, Euler’s name stands out, as well as Navier and Stokes for fluid mechanics equations, Fourier for heat equations, Maxwell for electromagnetism equations, and Schrödinger, Heisenberg, and Einstein for quantum mechanics and the theory of relativity PDEs, respectively (see e.g. [1, 6, 9] and the references therein). Nevertheless, the systematic examination of partial differential equations (PDEs) is relatively recent, with mathematicians embarking on this endeavor only in the twentieth century. A significant leap occurred with Schwartz’s formulation of the theory of distributions in the 1950s, and comparable progress emerged through Hörmander’s work on pseudo-differential calculus in the early 1970s. Importantly, the study of PDEs remains highly active as we progress into the twenty-first century [12–16]. Mathematics serves as a potent tool in both scientific inquiry and engineering applications, enabling precise modeling, analysis, and solution exploration of complex mathematical systems fundamental to advancing our understanding of the natural world and optimizing technological innovations [17–19, 21–23]. This research not only influences applied sciences but also plays a crucial role in the ongoing evolution of mathematics itself, particularly in the domains of geometry and analysis. In this work, the following problem is addressed:
in which \(\eta \geq 0\) for \(N=1,2\) and \(0<\eta \leq \frac{2}{N-2}\) for \(N\geq 3\), and \(h_{i}(.):R^{+}\rightarrow R^{+}\) (\(i=1,2\)) represents positive relaxation functions, which will be specified later. The term \(-\Delta ( . ) {tt}\) denotes the dispersion term, and \(M(\sigma )\) is a nonnegative locally Lipschitz function for \(\gamma ,\sigma \geq 0\) such that \(M(\sigma )=\alpha{1}+\alpha _{2}\sigma ^{\gamma}\). Specifically, we choose \(\alpha _{1}=\alpha _{2}=1\), and
In this context, we consider nonnegative constants \(\tau _{1}<\tau _{2}\) such that \(\beta {i} : [\tau{1}, \tau _{2}] \rightarrow \mathbb{R}\), where \(i=2,4\) represents the time delay in the distributive case. Furthermore, \(q(.)\), \(m(.)\), and \(s(.)\) are variable exponents defined as measurable functions on Ω̅ in the following manner:
where
with
and
This research is organized into distinct sections. In the following section, we present the hypotheses, concepts, and lemmas essential for our study. Section 2 is dedicated to proving the blow-up result, followed by the derivation of exponential growth of solutions. In Sect. 4, we establish the general decay when \(f_{1}=f_{2}=0\). The paper concludes with a comprehensive summary in the final section.
2 Fundamental theory
The importance of studying the blow-up of solutions in various systems lies in its ability to reveal critical thresholds, instabilities, and singularities that can significantly impact the behavior and evolution of dynamic processes [27–30]. Here, we will present some related theory and will define suitable assumptions for the proof of blow-up result.
(A1) Take a decreasing and differentiable function \(h_{i}: \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\) in a manner that
(A2) One can find \(\xi _{1},\xi _{2}>0\) in a way that
(A3) \(\beta _{i}:[\tau _{1}, \tau _{2}]\rightarrow \mathbb{R}\), \(i=2,4\), are a bounded functions satisfying
Lemma 2.1
There exists \(F(v, w)\) in a manner that
in which
Here, consider \(a_{1}=b_{1} = 1 \) for convenience.
Lemma 2.2
[26] One can find \(c_{0}>0\) and \(c_{1}>0\) in a way that
Consider a measurable function \(q:\Omega \rightarrow [1,\infty )\). We introduce the Lebesgue space with a variable exponent \(q(.)\) as follows:
with the norm defined by
Endowed with this norm, \(L^{q(.)}(\Omega )\) forms a Banach space. Subsequently, we introduce the variable-exponent Sobolev space \(W^{1,q(.)}(\Omega )\) as follows:
with the norm given by
\(W^{1,q(.)}(\Omega )\) is a Banach space, and the closure of \(C^{\infty}_{0}(\Omega )\) is given by \(W^{1,q(.)}_{0}(\Omega )\).
For \(v\in W^{1,q(.)}_{0}(\Omega )\), we give the equivalent norm
\(W^{-1,q'(.)}_{0}(\Omega )\) sign to the dual of \(W^{1,q(.)}_{0}(\Omega )\) in which \(\frac{1}{q(.)}+\frac{1}{q'(.)}=1\).
Also, we take the log-Hölder inequality
for all \(y,z\in \Omega \), with \(\vert y-z\vert <\zeta \), where \(0<\zeta <1\) and \(A>0\).
Theorem 2.3
Assume (2.1)–(2.3) hold. Then, for any \((v_{0},v_{1},w_{0},w_{1},f_{0},g_{0})\in \mathcal{H}\), (1.1) has a unique solution for some \(T>0\):
where
Proof
We can prove the local existence result for (1.1) in suitable Sobolev spaces by exploiting the Faedo–Galerkin approximation method (see [3, 24]). □
Firstly, we take the following variables as mentioned in [25]:
which verify
and
Then, problem (1.1) is equivalent to
where
In the upcoming step, the energy functional is introduced.
Lemma 2.4
Let (2.1)–(2.3) be satisfied, and assume that \((v,w,x,z)\) is a solution of (2.8), then \(E(t)\) is nonincreasing, that is,
fulfills
where
Proof
By multiplying (2.8)1, (2.8)2 by \(v_{t}\), \(w_{t}\) and integrating over Ω, we have
Now, multiplying (2.8)3 by \((\frac{\delta m(y)-1}{m(y)} \vert x(y,1,r,t) \vert ^{m(y)-1} \vert \beta _{2}(r)\vert )\), then integrating over \(\Omega \times (0, 1)\times (\tau _{1}, \tau _{2})\), and applying (2.6)2, the following is obtained:
and by the inequalities of Young, we have
Hence,
Similarly, we get
and
According to (2.12), (2.13), (2.15), (2.16), (2.17), we find (2.9) and
Hence, by (2.3), we obtain (2.10), where
and hence E is a decreasing function, which completes the proof. □
3 Blow-up
Here, we establish the blow-up result for the solution of (2.8). Initially, we introduce the functional as follows:
Theorem 3.1
Assume that (2.1)–(2.3) hold and assume \(E(0)<0\), then the solution of (2.8) blows up in finite time.
Proof
From (2.9), the following can be written:
Therefore
Hence
in which
Lemma 3.2
Let \(\exists c>0\) in a way that any solution of (2.8) fulfills
Proof
Let
we have
then
Hence, we get
According to (3.5), we have
Therefore,
Hence
Similarly, we find
The adding of (3.11) and (3.12) gives us (3.6). □
Corollary 3.3
Proof
From (1.5), we have
According to Lemma 3.2, we find (3.13)1. Similarly, we obtain (3.13)2. □
Now, take
in which \(0<\varepsilon \) will be considered later and take
By multiplying (2.8)1, (2.8)2 by v, w and with the help of (4.4), the following is achieved:
By (2.1), we obtain
We have
From (4.5), we find
Applying the inequality of Young, we have for \(\delta _{1},\delta _{2}>0\)
and
Therefore, by setting \(\delta _{1}\), \(\delta _{2}\) so that
putting in (3.20), the following is obtained:
in which \(\widehat{m}=\frac{m^{+}-1}{m^{-}}\), \(\widehat{s}=\frac{s^{+}-1}{s^{-}}\), by using (3.5) and (3.13), we have
By (3.16), we find
and by the inequality
with \(b=\frac{1}{\mathbb{H}(0)}\). Then we have
and
where \(C_{3}=1+\frac{1}{\mathbb{H}(0)}\). Substituting (3.29) and (3.30) into (3.27), we get
Similarly, we find
where \(C_{4}=C_{4}(\kappa )=C_{3}\frac{\beta _{1}(\delta +1)}{\delta m^{-}}( \frac{C_{0}\kappa}{2})^{1-m^{-}}\), \(C_{5}=C_{5}(\kappa )=C_{3}\frac{\beta _{3}(\delta +1)}{\delta s^{-}}( \frac{C_{0}\kappa}{2})^{1-s^{-}}\).
Combining (3.31), (3.32), and (3.26), and by (2.4), we obtain
Now, for \(0< a<1\), from (3.1) and (2.4)
Substituting (4.21) in (4.20) and applying (2.4), the following is obtained:
Here, choose \(0< a\) in a manner that
Further, we have
and suppose
which gives
After that, select κ large enough that
In the last stage, take κ, a, and we pick ε in a way that
and
Thus, for some \(\mu >0\), (3.35) implies
and
In the coming step, applying the inequalities of Holder and Young, we get
where \(\frac{1}{\mu}+\frac{1}{\theta}=1\).
Select \(\mu =(\eta +2)(1-\alpha )\) to obtain the following:
Consequently, by the application of (3.5), (3.16), and (3.28), we have
Therefore, we have
In the same way, we have
where \(\frac{1}{\mu}+\frac{1}{\theta}=1\).
For the next step, assume \(\theta =2(\gamma +1)(1-\alpha )\) to get
in which \(0< \lambda \), this relies only on β and c.
Further simplification of (4.31) leads to
Hence, \(\mathcal{K}(t)\) blows up in time
Thus, it completes the proof. □
4 Growth of solution
Here, the exponential growth of solution of problem (2.8) will be established.
For this, the functional is defined as follows:
Theorem 4.1
Assume that (2.1)–(2.3) are satisfied, and suppose \(E(0)<0\), then
Then the solution of problem (2.8) grows exponentially.
Proof
To prove the required result, (2.9) implies
with the help of (3.3) and (3.4).
Now, take the following:
in which \(\varepsilon >0\) will be chosen in a later stage.
From (2.8)1, (2.8)2, and (4.4), we have
By (2.1), we obtain
Similar to \(J_{1}\), \(J_{2}\) in (3.21) and (3.22), we estimate \(I_{1}\), \(I_{2}\):
From (4.5), we find
Similar to \(J_{3}\), \(J_{4}\), \(J_{5}\), and \(J_{6}\) in (3.20)–(3.24), we estimate \(I_{i}\), \(i=3,\ldots,6\). By Young’s inequality, we find for \(\delta _{1},\delta _{2}>0\)
and
Therefore, by setting \(\delta _{1}\), \(\delta _{2}\) so that
substituting in (4.8), the following is achieved:
where \(\widehat{m}=\frac{m^{+}-1}{m^{-}}\), \(\widehat{s}=\frac{s^{+}-1}{s^{-}}\). By using (3.5) and (3.13), we have
By (1.5), we find
and by (3.28) with \(b=\frac{1}{\mathbb{H}(0)}\). Then we have
and
where \(C_{10}=1+\frac{1}{\mathbb{H}(0)}\). Substituting (4.16) and (4.17) into (4.15), we get
Similarly, we find
where \(C_{11}=C_{11}(\kappa )=C_{9} \frac{\beta _{1}(\delta +1)}{\delta m^{-}}(\frac{C_{0}\kappa}{2})^{1-m^{-}}\), \(C_{12}=C_{12}(\kappa )=C_{9} \frac{\beta _{3}(\delta +1)}{\delta s^{-}}(\frac{C_{0}\kappa}{2})^{1-s^{-}}\).
Combining (4.18), (4.19), and (4.14), we have
Now, for \(0< a<1\), from (4.1) and (2.4)
Substituting (4.21) in (4.20) and applying (2.4), we have
Here, assume that \(0< a\) is small in a manner that
we have
and we assume
which gives
After this, select κ large in a way that
At the last stage, fix κ, a and pick ε small such that
and
and from (4.4)
Thus, for some \(\mu _{1}>0\), (4.22) implies
and
Further, applying the inequalities of Holder and Young, we get
where \(\frac{1}{\mu}+\frac{1}{\theta}=1\). Next, assume \(\mu =(\eta +2)\) to reach
Subsequently, by using (4.2) and (3.28), we obtain
Therefore
Hence
From (4.26) and (4.30), we have
where \(\lambda _{1}> 0 \), this relies on \(\mu _{1} \) and c only. Further, (4.31) implies
Therefore, we deduce that the solution experiences exponential growth in the \(L^{2(p^{+}+2)}\) norm. This concludes the proof. □
5 General decay
In this section, we state and prove the general decay of system (2.8) in the case \(f_{1}=f_{2}=0\). For this goal, problem (2.8) can be written as
where
Here, we introduce the modified energy functional \(\mathcal{E}\) of (5.1) as follows:
Similar to Lemma 2.4, the energy functional fulfills for assumption (2.3)
where
Remark 5.1
In this case \(f_{1}=f_{2}=0\). Condition (2.3) remains true for (\(\delta =1\)), i.e., it can be replaced by
Also, relation (5.3) becomes of the form
Lemma 5.2
(Komornik, [20]) Assume a nonincreasing function \(E:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\), and suppose that \(\exists \sigma ,\omega >0\) in a manner that
Then we have \(\forall t\geq 0\)
Theorem 5.3
Assume that (1.3), (2.1)–(2.3), and (2.5) hold. Then \(\exists c,\lambda >0\) such that the solution of (5.1) fulfills
Proof
Multiplying (5.1)1 by \(v \mathcal{E}^{p}(t)\) for \(p>0\) to be specified later and integrating the result over \(\Omega \times (s,T), s< T\), we have
which implies that
By (5.2) and the relation
this implies
At this point, we estimate \(I_{i}\), \(i=1,\ldots,12\), of the RHS in (5.12), we have
Since \(\mathcal{E}\) is decreasing, this implies
Similarly, we find
and
Next, we get
The other terms are estimated as follows:
For the next term, we have
and applying the inequality of Young, the following is obtained:
Here, utilizing \(H_{0}^{1}(\Omega )\hookrightarrow L^{m^{-}}(\Omega )\) and \(H_{0}^{1}(\Omega )\hookrightarrow L^{m^{+}}(\Omega )\), we get
Similarly, we find
and
Now, by the inequality of Young from the last term, the following is obtained:
By substituting (5.14)–(5.25) into (5.12), we find
Now, choose ε so small that
After, fix ε, \(c_{\varepsilon}(y)\leq M\) because \(m(y)\) is bounded.
Hence, by (5.3),
Taking \(T\rightarrow \infty \), we get
Finally, Komornik’s Lemma 5.2 (with \(\sigma =p=\frac{m^{+}-2}{2}\)) implies our result. This completes the proof. □
6 Conclusion
In this research, we investigated the blow-up and growth of solutions in a coupled nonlinear viscoelastic Kirchhoff-type system with sources, distributed delay, and variable exponents. Additionally, we obtained a general decay result when \(f_{1}=f_{2}=0\) by leveraging an integral inequality introduced by Komornik [20]. Such problems are commonly encountered in various mathematical models of real-world problems. In future research, we plan to apply this approach to address similar problems, incorporating additional damping effects such as Balakrishnan–Taylor damping and logarithmic terms. We will also try to prove the general decay result in the case (\(f_{1},f_{2}\neq 0\)).
Data Availability
No datasets were generated or analysed during the current study.
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Boulaaras, S., Choucha, A., Ouchenane, D. et al. Blow up, growth, and decay of solutions for a class of coupled nonlinear viscoelastic Kirchhoff equations with distributed delay and variable exponents. J Inequal Appl 2024, 55 (2024). https://doi.org/10.1186/s13660-024-03132-2
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DOI: https://doi.org/10.1186/s13660-024-03132-2