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Blow-up of solutions for a system of nonlocal singular viscoelastic equations with sources and distributed delay terms
Boundary Value Problems volume 2024, Article number: 77 (2024)
Abstract
In this paper, we investigate a scenario concerning a coupled nonlocal singular viscoelastic equation with sources and distributed delay terms. By establishing suitable conditions, we have proved that a finite-time blow-up occurs in the solution.
1 Introduction
In this paper, we examine the subsequent system comprising two singular nonlinear and nonlocal equations, inspired by a one-dimensional viscoelastic system:
where
and \(Q=(0,L)\times (0,T)\times (\tau _{1},\tau _{2}))\), \(L<\infty \), \(T<\infty \), \(\mu _{1},\mu _{3} >0\), \(g_{1}(.)\), \(g_{2}(.): \mathbb{R} ^{+}\rightarrow \mathbb{R} ^{+}\) and \(f_{1},f_{2}:\mathbb{R}^{2}\longrightarrow \mathbb{R}\) are functions stated in (8).
These issues arise in one-dimensional or longitudinal elasticity when considering long-term memory viscosity. The second integral represents the distributed delay terms, where \(\tau _{1}, \tau _{2} >0\) denote time delays, \(\mu _{2}\), \(\mu _{4}\) are \(L^{\infty}\) functions.
Our work is motivated by the findings presented in the following papers:
In [13], the authors examined a model depicting the motion of a viscoelastic two-dimensional body on the unit disc, focusing on radial solutions. Furthermore, they established the uniqueness and existence of the generalized solution for the below stated nonlocal problem
where \(Q=(0,1)\times ( 0,T ) \) with right hand side f is a Lipshitzian function.
The findings presented in [14] indicate the occurrence of blow-up for large initial data and demonstrate decay outcomes for sufficiently small initial data, which are applicable to the subsequent nonlocal singular problem
They acquired the blow-up properties of local solutions utilizing Georgiev–Todorova method, even with negative initial energy. Authors in [11], employed the direct method to prove blow-up of solutions under appropriate conditions on initial data [23]. The authors in [26], extended the previous result to systems with higher dimensions and get additional blow-up results. The authors in [15], substituted the source terms \(f_{1}(u,v)\) and \(f_{2}(u,v)\) in the studied system in [26], respectively by \(\vert v \vert ^{q+1} \vert u \vert ^{p-1}u\), \(\vert u \vert ^{p+1} \vert v \vert ^{q-1}v\) and the Bessel operator \(\frac {1}{x}\frac {\partial }{\partial x} ( x \frac {\partial }{\partial x} ) \) instead of Laplace operator Δ and considered the nonlocal boundary condition
and with two different functions \(g(.)\). Moreover, it is augmented by both nonlocal and classical condition. Furthermore, in [17], the authors investigated the identical problem presented in [15], where they derived a nonlinear source of polynomial nature. This source is capable of inducing solutions to blow up within a finite time frame, even in the existence of enhanced damping \(u_{t}\). They considered three different cases regarding the sign of the initial energy. PiÅŸkin and Ekinci [22] have addressed problem (1) by substituting Bessel operator with the Kirchhoff operator featuring degenerate damping terms. They employed a technique identical to that used to establish the global existence and provide a decay rate for solutions, as well as demonstrate a finite-time blow-up when the behavior of decreasing relaxation functions is stated as:
and \(\xi (t)\) satisfies
Boulaaras et al. [18] investigated the subsequent system, which consists of two singular one-dimensional nonlinear equations arising in generalized viscoelasticity, featuring long-term memory, nonlocal boundary conditions, and general source terms
By employing potential-well theory the authors proved the existence of a global solution for the problem. And in the same vein, in [5] Boularaas and Mezouar proved the existence and decay of solutions of a singular nonlocal viscoelastic system featuring nonlocal boundary conditions, localized damping term and linear source term. In domain of blow-up phenomena, the authors in [27] investigated the finite-time blow-up of solutions for an initial boundary value problem with nonlocal boundary conditions, pertaining to a system of nonlinear singular viscoelastic equations. Other works in the same vein can be found here [1–4, 7, 10–12, 24, 25].
The influence of delay frequently emerges in numerous applications and practical issues, transforming various systems into distinct problems warranting investigation. Recently, numerous authors have examined asymptotic behavior, stability, and blow-up phenomena of solutions in evolution systems with time delay. Refer to works by [6, 8, 9, 19] for further details.
Motivated by the aforementioned works, in this study, we expand upon the earlier investigation outlined in [15] to encompass singular one-dimensional nonlinear viscoelastic equations with source and distributed delay terms. Specifically, we delve into the blow-up phenomenon of solutions with negative initial energy for problem (1).
In the following, let \(c,c_{i},C>0\), are positive constants.
Our paper is structured as follows: In the subsequent section, we establish concepts, lemmas, and hypotheses essential for our analysis. In Sect. 3, we state and prove the blow-up phenomenon of solutions.
2 Preliminaries
In this section, we present the following definitions, symbols, spaces and lemmas that we utilize throughout the paper.
Let \(L_{x}^{p}=L_{x}^{p}(0,L)\) represent the weighted Banach space having norm
Hilbert space of square integral functions is denoted by \(H=L_{x}^{2}(0,L)\) having the finite norm
Hilbert space is represented by \(V=V_{x}^{1}((0,L)) \) equipped with the norm
and
Lemma 1
(Poincare-type inequality) For any v in \(V_{0}\), we have
and
Remark 1
Clearly \(\Vert u \Vert _{V_{0}}= \Vert u_{x} \Vert _{H}\) defines an equivalent norm on \(V_{0}\).
Theorem 1
(See [1]) For any v in \(V_{0}\) and \(2< p<4\), we have
where \(C_{\ast }\) is a constant depending on L and p only.
We demonstrate the blow-up outcome given the following appropriate assumptions.
(A1) \(g_{1},g_{2}: \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\) are decreasing and differentiable functions such that
(A2) There exists a constants \(\xi _{1},\xi _{2}>0\) such that
(A3) \(\mu _{2},\mu _{4}:[\tau _{1}, \tau _{2}]\rightarrow \mathbb{R}\) is a bounded function satisfying
By combining arguments of other studies [16, 20, 22] with used the Faedo–Galerkin method, we get the local existence theorem.
Theorem 2
Let (3), (4), and (5) hold. Assume that
Then, for any \((u_{0},v_{0})\in V_{0}^{2}\), \((v_{1},v_{2})\in H^{2}\) and \((h_{0},k_{0})\in \mathcal{H,}\) problem (1) has a unique local solution
for \(T^{*}>0\) small enough, where \(\mathcal{H}=L^{2}_{x}((0,L)\times (0,1)\times (\tau _{1},\tau _{2}))\).
Lemma 2
There exists a function \(F(u, v)\) such that
where
We take \(a_{1}=b_{1} = 1 \) for convenience.
Lemma 3
[21] There exist two positive constants \(c_{0}\) and \(c_{1}\) such that
Taking new variable as in [19],
so we get
Consequently, the problem (1) is equivalent to
We define the energy functional.
Lemma 4
Assume (3), (4), (5), and (6) hold, let \((u,v)\) be a solution of (1), then \(E(t)\) is non-increasing, that is
satisfies
where
and
Proof
By multiplying (1)1, (1)2 by \(xu_{t}\), \(xv_{t}\), respectively, and integrating over \((0,L)\), we get
Now, multiplying the Equation (8)3 by \(xz\vert \mu _{2}(s)\vert \) and integrating the result over \((0, L)\times (0, 1)\times (\tau _{1}, \tau _{2})\)
using Young’s inequality, we achieve
Similarly, we get
and
By remplacing (13)–(16) into (12), we obtain (9) and
Hence, according to (4) and (5), we get (10), where \(C_{0}=\min \{\mu _{1}-(\delta +\frac{1}{2})\int _{\tau _{1}}^{ \tau _{2}}\vert \mu _{2}(s)\vert \,ds,\mu _{3}-(\delta +\frac{1}{2}) \int _{\tau _{1}}^{\tau _{2}}\vert \mu _{4}(s)\vert \,ds , \frac{2\delta -1}{4\delta} \}>0\).
The proof is completed. □
3 Blow-up
In this segment, we establish the blow-up outcome for the solution of problem (1).
Following this, we introduce the functional
Theorem 3
Let (3)–(5), and (6) holds and \(E(0)<0\), then solution of problem (1) blow-up in finite time.
Proof
From (10), we have
Therefore
this implies that
We set
where
By multiplying (1)1, (1)2 by xu, xv and derivative of (23), we achieve
We have
Next, by Young’s inequality, we find for \(\delta _{1}>0\)
and
from (25), we get
At this stage, we choose \(\delta _{1}\) so that, for large κ to be chosen later
by (21) and putting in (30), we get
For \(0< a<1\) and from (18), we obtain
Substituting in (31) and by (7), we achieve
According to (22), we get
by using the fact that
we have
where \(d=(1+\frac{1}{\mathcal{H}(0)}\).
On the other hand, using the following inequality
we have
from (24), we get
Similarly, we find
Substituting (35) and (36) into (33), we get
where \(c_{4}=\min \{c_{2}\mu _{1},c_{3}\mu _{3}\}\).
Now, taking \(a>0\) small enough such that
let
gives
Next, we pick κ in a way that
Chose fixed values for κ and a and select ε small enough such that
and
Thus, for some \(\beta _{1}>0\), estimate (37) becomes
and
Now, utilizing Young’s and Holder’s inequalities, we obtain
where \(\frac{1}{\mu}+\frac{1}{\theta}=1\).
We chose \(\theta =2(1-\alpha )\) to achieve
For \(s=\frac{2}{(1-2\alpha )}\) and by using (24) and (34), we obtain
Therefore,
Next,
where \(\lambda > 0 \), which depends on c and \(\beta _{1}\) only.
By integrating (44), we get
Thus, the solution blows up in a finite time \(T^{*}\), with
Then the proof is complete. □
Data Availability
No datasets were generated or analysed during the current study.
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Choucha, A., Shahrouzi, M., Jan, R. et al. Blow-up of solutions for a system of nonlocal singular viscoelastic equations with sources and distributed delay terms. Bound Value Probl 2024, 77 (2024). https://doi.org/10.1186/s13661-024-01888-6
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DOI: https://doi.org/10.1186/s13661-024-01888-6