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Blow-up and lifespan of solutions for elastic membrane equation with distributed delay and logarithmic nonlinearity
Boundary Value Problems volume 2024, Article number: 36 (2024)
Abstract
We examine a Kirchhoff-type equation with nonlinear viscoelastic properties, characterized by distributed delay, logarithmic nonlinearity, and Balakrishnan–Taylor damping terms (elastic membrane equation). Under appropriate hypotheses, we establish the occurrence of solution blow-up.
1 Introduction
Analyzing nonlinear mathematical problems involves a distinct set of challenges and techniques compared to linear problems [1, 2]. Nonlinear problems often manifest in various scientific and engineering domains, and their analysis is crucial for gaining insights into complex phenomena [3, 4]. The systematic examination of mathematical problems necessitates a methodical approach encompassing rigorous formulation, assessment of existence and uniqueness of solutions, linearization procedures, stability analyses, application of numerical methodologies, bifurcation investigations, phase plane analyses, sensitivity assessments, optimization strategies, and the validation and verification of outcomes [5, 6]. This multifaceted framework is imperative for elucidating the intricate dynamics inherent in various systems across diverse scientific and engineering domains [7].
The analysis of solutions to the Kirchhoff equation concerning viscoelastic materials is paramount in comprehending the mechanical characteristics of such materials. These solutions serve as a cornerstone in directing the design methodology, thereby ensuring the dependability and efficacy of materials across diverse applications. In this current study, we examine the Kirchhoff equation provided below:
where
In this expression, Ω belongs to the set of bounded domains in \(\mathbb{R}^{N}\) and possesses a boundary ∂Ω that is suitably smooth. \(\gamma \geq 2\), \(\zeta _{0}\), \(\zeta _{1}\), σ, \(\mu _{1}\), l are positive constants. In addition to this, the time delays are indicated by \(\tau _{1}\), \(\tau _{2}\) with \(0\leq \tau _{1}<\tau _{2}\), while \(\mu _{2}\) is an \(M^{\infty}\) function and f is a positive function.
In a physical sense, the connection between the stress and strain history in the beam is influenced by a viscoelastic damping term inspired by Boltzmann theory. The kernel of memory term in this context is represented by the function f, which is frequently discussed in the literature [8–16].
In [17], Balakrishnan and Taylor introduced a novel damping model known as Balakrishnan–Taylor damping, specifically addressing concerns related to the span problem and the plate equation. Numerous studies have explored this damping phenomenon, as documented in [11, 14, 15, 17–20, 35–37], and [21]. The occurrence of delay is a common feature in various applications and practical problems, rendering many systems worthy of investigation. Recently, several authors have directed their attention towards analyzing the asymptotic behavior and stability of evolution systems incorporating time delay, as discussed in the research [9–12, 15, 22–26], and [27].
The significance of logarithmic nonlinearity in physical system is emphasized by its involvement in a wide range of topics and theories, encompassing symmetry, cosmology, quantum mechanics, nuclear physics, and various applications including nuclear, optical, and subterranean physics. Different authors have delved into such type of problem across various domains, exploring aspects such as global solution existence, stability, blow-up, and the growth of solutions, as documented in works such as [11, 19, 28–31], and [32–34]. Taking into account various elements, damping terms including the distributed delay terms, logarithmic nonlinearity, Balakrishnan–Taylor damping, and memory term are integrated into a specific problem, along with the incorporation of \(\int _{\tau {1}}^{\tau {2}}\mu{2} (s)v{t}(y,t-s) \,ds\). Further investigation is required to explore such type of novel and distinctive problem. This diverges from the previously mentioned scenarios, and our objective is to shed light on this unique problem.
Our work is structured as follows: In the subsequent section, we lay out the necessary lemmas, concepts, and hypotheses. In Sect. 3, we state and prove the main blow-up solution results. We present the concluding remarks of our work in Sect. 4 of this work.
2 Fundamental concepts
Here, to investigate our problem, we require certain materials. To begin with, we present the following assumptions regarding \(\beta _{2}\) and f:
- (H1):
-
\(f:\mathbb{R}{+}\rightarrow \mathbb{R}{+}\) represents nonincreasing \(B^{1}\) functions fulfilling
$$ 0< f(t), \quad \zeta _{0}- \int _{0}^{\infty}f( \Lambda ) \,d\Lambda = l>0. $$(2.1) - (H2):
-
\(\mu _{2}:[\tau _{1},\tau _{2}]\rightarrow \mathbb{R}\) is an \(M^{\infty}\) function in a way that
$$ \biggl(\frac{2\delta +1}{2}\biggr) \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2}(s) \bigr\vert \,ds< \mu _{1} , \quad \delta >\frac{1}{2}, $$(2.2)with
$$ (f\circ \psi ) (t):= \int _{\Omega} \int _{0}^{t}f(t-\Lambda ) \bigl\vert \psi (t)- \psi (\Lambda ) \bigr\vert ^{2}\,d\Lambda \,dy. $$
As in [27], we take the following:
which satisfy
Then one can write (1.1) as follows:
Here, the energy functional is introduced as follows.
Lemma 2.1
Let Q represent the energy functional given by
which satisfies
Proof
By taking the inner product of (2.4)1 with vt and subsequently integrating over Ω, we obtain
Then a straightforward calculation yields
further simplification implies that
with
and
Then we have
Using (2.1), one has
and
By substituting (2.13) and (2.14) into (2.12), we have
and
Here, multiply \(x\vert \mu _{2}(s)\vert \) with equation (2.4)2 by \(x\vert \mu _{2}(s)\vert \) and integrate over \(\Omega \times (0, 1)\times (\tau _{1},\tau _{2})\). Then applying (2.3)2, the following is obtained:
and through the application of Young’s inequality, we obtain
By replacement (2.8)–(2.9) and (2.15)–(2.18) into (2.7), we find (2.5) and
Thus, according to (2.2), we get(2.6), where \(B_{0}>0\). Hence, the proof is completed. □
Theorem 2.2
Let us consider that (2.1)–(2.2) hold true. For any \(v_{0},u_{1}\in F^{1}{0}(\Omega )\cap M^{2}(\Omega )\) and \(f{0}\in M^{2}(\Omega ,(0,1))\), one can find a weak solution v to (2.4) such that
Lemma 2.3
[34] One can find a constant \(b(\Omega )>0\) in a manner that
for any \(2\leq s\leq \gamma \), provided that \(0 \leq \int _{\Omega}\vert v\vert ^{\gamma}\ln \vert v\vert ^{l} \,dy\).
Corollary 2.4
[34] One can find a constant \(b(\Omega )>0\) in a way that
provided that \(0 \leq \int _{\Omega}\vert v\vert ^{\gamma}\ln \vert v\vert ^{l} \,dy\).
Lemma 2.5
[34] Let us take a constant \(b(\Omega )>0\) in a way that
for any \(v\in M^{\gamma}(\Omega )\) and \(2\leq s\leq \gamma \).
3 Blow-up result
Here, we establish the blow-up results for the solution of (2.4). First of all, the functional is introduced as
Theorem 3.1
Assuming that (2.1)–(2.2) are satisfied, and given that \(Q(0)<0\), the solution to problem (2.4) experiences a finite time blow-up.
Proof
For the required proof, the following is obtained from (2.6):
thus, we have
which implies that
By (3.1), we have
We set
where \(\varepsilon >0\) will be assigned a specific value later, and
Multiplying v with (2.4)1 and taking the derivative of (3.6), the following is obtained:
Next, we have
and, for \(\delta _{1}>0\),
From (3.8), we find
At this point, by setting \(\delta _{1}\) so that, for large κ to be specified later
by (3.4) and putting in (3.11), we get
Now, for \(0< a<1\) and from (3.1), we have
Putting in (3.12), one has
According to (3.5), Corollary 2.4, and Young’s inequality, we get
(3.7) yields
Hence, Lemma 2.3 gives
From (3.14) and (3.15), we have
At this point, we take \(a>0\) small enough so that
and we assume that
gives
then we select κ in a way that
Finally, we set a and κ as fixed values and select ε to be sufficiently small fulfilling
and
This implies that for some \(\eta >0\) estimate (3.14) becomes
Subsequently, employing the inequalities of Holder and Young, we obtain
where \(\frac{1}{\mu}+\frac{1}{\theta}=1\). We take \(\mu =2(1-\alpha )\) to get
Further, for \(s = \frac{2}{2(1-\alpha )-1}\), estimate (3.19) gives
Then, Lemma 2.5 yields
Hence,
Next, (3.18) and (3.21) implies
with \(\Gamma > 0 \), this relies on η and b only.
By integration of (3.22), we have
Hence, \(\mathcal{L}(t)\) blows up in time
The proof is completed. □
4 Conclusion
The examination of solutions to the Kirchhoff equation in the context of viscoelastic materials holds paramount significance. In our investigation, a Kirchhoff-type equation featuring nonlinear viscoelastic properties, distinguished by distributed delay, logarithmic nonlinearity, and Balakrishnan–Taylor damping terms, was examined. Subsequent to verifying pertinent hypotheses, the manifestation of solution blow-up was conclusively established.
Data Availability
No datasets were generated or analysed during the current study.
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SB: writing original draft, Methodology, RJ: Resources, Methodology, AC: formal analysis, Conceptualization; AZ, MB conceptualized, investigated, analyzed and validated the research while; SB: formulated, investigated, reviewed and, Corresponding author, Supervision.
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Boulaaras, S., Jan, R., Choucha, A. et al. Blow-up and lifespan of solutions for elastic membrane equation with distributed delay and logarithmic nonlinearity. Bound Value Probl 2024, 36 (2024). https://doi.org/10.1186/s13661-024-01843-5
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DOI: https://doi.org/10.1186/s13661-024-01843-5