Blow up of solutions for a system of two singular nonlocal viscoelastic equations with damping, general source terms and a wide class of relaxation functions

1.   Introduction

Mixed non local problems for hyperbolic and parabolic PDEs, have been studied intensively in recent decades. Such equations or systems with constraints modelize many time-dependant physical phenomena. These constraints can be a data measured directly on the boundary or giving integral boundary conditions (se for example ^{[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]}).

The aim of this work, is to show the blow up of solutions of the viscoelastic one-dimensional system

$f_{1}(., .)$, $f_{2}(., .):\mathbb{R}^{2}\longrightarrow \mathbb{R}$ are given by

with $r\geq -1$, $a_1, b_1 \in \mathbb{R}$.

$D_T = (0, L)\times (0, T), \; L, T, \mu_{1}$ and $\mu_{2}$ are positive constants, $g_{1}, g_{2}: \mathbb{R} ^{+}\rightarrow \mathbb{R} ^{+}$ are functions to be specified later.

This work is motivated by ^[11], where S. Mesloub studied a problem modelizing the movement of a two-dimensional viscoelastic object on a disc :

where $D_T = \{ \left(x, t\right) \in \mathbb{R}^{2}, \ 0 < x < l, \ 0 < t < T\}$, and the source term $f$ verifies some Lipschitz conditions.

Using an iterative process, he proved the existence and uniqueness of the solution of the nonlinear problem (1.3).

Later in ^[14], S.A. Messaoudi showed the existence of solutions with positive initial energy that blow up in finite time of the following nonlinear viscoelastic hyperbolic problem

$\Omega$ is a bounded domain of $\mathbb{R}^{n}$, $(n\geq 1)$ with a smooth boundary $\partial \Omega$, $p > 2$, $m\geq 1$.

More work followed up on similar nonlocal singular viscoelastic equations and systems in ^{[4,7,8,15,20,21]}.

Recently in ^[3], we prove global existence and decay for system 1.1, by constructing a Lyapunov function combined with a perturbed energy.

In ^[1] with absence of the damping term ($\mu_{1} = \mu_{2} = 0$), the authors established a general decay result to the system 1.1.

In this work, we continue our study on system 1.1. We start by giving the fundamental definitions and theorems on function spaces that we need, then we state the local existence theorem. Finally, we state and prove with suitable conditions the blow up in finite time of solutions for system 1.1.

2.   Preliminaries

We start by defining the weighted Banach space $L_{x}^{p} = L_{x}^{p}((0, L))$ equipped with the norm

For $p = 2$, we obtain the Hilbert space $H = L_{x}^{2}((0, L))$ equipped the finite norm

Consider also the Hilbert space $V : = V_{x}^{1}((0, L))\ $ with finite norm

and

As in Sobolev spaces, one can prove the following lemma

Lemma 1. There exists $C > 0$ such that

Remark 1. Observe that in $V_{0}$, $\left\Vert u\right\Vert_{V}$ is equivalent to $\left\Vert u_{x}\right\Vert_{H}$.

Theorem 1. (^[2]) There exists constant $C_p > 0$ depending only on $L$ and $p$, such that for $2 < p < 4,$ and any $v$ in $V_{0}$ we have

Before stating the local existence of solutions result, we consider the following assumptions

(A1) $g_{1}, g_{2}: \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ are two differentiable and decreasing functions with

(A2)There exist non-increasing differentiable functions $\vartheta_{1}, \vartheta_{2} : [0, +\infty)\rightarrow(0, +\infty)$ and $C^{1}$ functions $\Phi_{1}, \Phi_{2} : [0, +\infty)\rightarrow[0, +\infty)$ which are linear or strictly increasing and strictly convex $C^{2}$ functions on $(0, \varepsilon], \varepsilon\leq g_{i}(0)$, with $\Phi_{i}(0) = \Phi'_{i}(0) = 0$ such that

(A3) $r > -1$.

First we have to introduce the definition of a weak solution to (1.1).

Definition 1. We say that dualism $(u, v)$ is a weak solution to the system (1.1) on $[0, T]$ if

In addition, $(u, v)$ satisfies

for all test function $\phi, \varphi \in V_{0}(0, L),$ and for a.e. $t \in[0, T]$

Theorem 2. Assume (A1)–(A3) hold.

Then, there exists a small enough positive number $T^*$ such that system (1.1) admits a unique local solution $\left(u, v\right) \in \bigg[C((0, T^{*}); V_{0})\cap C^{1}((0, T^{*}); H)\bigg]^{2},$

$\forall (u_{0}, v_{0})\in V_{0}^{2}, \ \ \forall (u_{1}, v_{1})\in H^{2}$.

Remark 2. The proof of this theorem can be established exactly as in ^[22], and ^[3] where we also proved a global existence result for problem (1.1).

Lemma 2. Let $F(u, v)$ be a function defined as follows

Then

and

Take $a_{1} = b_{1} = 1$ for convenience.

Lemma 3. ^[17] There exist $c_{1}$, $c_{2}$ positive constants such that

Lemma 4. Assume (A1), (A2) and (A3) hold, and $(u, v)$ be a solution of (1.1), then the energy functional

is non-increasing and it satisfies

where

and

Proof. By integration by parts, we obtain

By multiplying the first and second equations of system (1.1) by $xu_{t}, \quad xv_{t}$ respectively, and integrating over $(0, L)$, then using (2.14)–(2.19), we get

From (2.7), (2.8) and (2.20), we obtain (2.11).

Lemma 5. ^[16] There exist positive constants $d$ and $t_{0}$ such that, for any $t\in[0, t_{0}]$, we have

Lemma 6. If (2.7) hold. Then, for any $\phi\in V_{0}$, $0 < \alpha < 1$ and $i = 1, 2$, we have

where $C_{\alpha, i}: = \int_{0}^{\infty}\frac{g_{i}^{2}(s)}{\alpha g_{i}(s)-g_{i}'(s)}ds$ and $h_{i}: = \alpha g_{i}-g_{i}'$.

3.   Blow up

Now, we give the main result of this paper

Theorem 3. Assume (A1)–(A3) hold, and $E(0) < 0$. Then the solution of problem (1.1) blows up in finite time.

Proof. Let us define the functional $I$ by

From (2.11) and the assumption $E(0) < 0$, we get

and

By (2.13), (2.9) and (3.3) we have

Set

with

By multiplying the first and the second equations of system (1.1) by $xu, xv$, we can verify that the derivative of (3.5) is given by

we have

and

So, by (3.7)

For $0 < \lambda < 1$, from (3.1)

\label{sys2.23} We obtain by substituting in (3.10)

by (2.22), we find

where $C_{\alpha} = \min\{C_{\alpha, 1}, C_{\alpha, 2}\}$.

Now, one cause the Lebesgue dominated convergence theorem with the fact that $\frac{g_{i}^{2}(s)}{\alpha g_{i}(s)-g_{i}'(s)} < g_{i}(s)$, for $i = 1, 2$, to prove that

Therefore, there exists $\alpha_{0} \in(0, 1)$ such that if $\alpha < \alpha_{0}$, then, by letting $\alpha = \varepsilon(r+2)(1-\lambda) < \alpha_{0}$, we get

We put $\lambda > 0$ small enough, we have

at this point, we take

then, we have

i.e $\beta_{2} > 0.$

Similarly

Choosing $\varepsilon$ small enough, thus we have

Then, for some $\gamma_{1}, \gamma_{2} > 0$, estimate (3.12) becomes

By (2.9) and (2.11), for some $\gamma_{3} > 0$, we get

We obtain,

where $\mathcal{J}_{1}(t): = \mathcal{J}(t)+\gamma_{2}I(t)$,

and

On the other hand, we have by Holder's and Young's inequalities,

where $\frac{1}{\nu}+\frac{1}{\mu} = 1$.

Taking $\mu = 2(1-\delta)$, we get

Then, for $s = \frac{2}{(1-2\delta)}$ and from (3.1), we obtain

So, for some $k_2 > 0$

Therefore, there exist $k_3 > 0$ such that

From (3.15) and (3.21), we finally get the required inequality

where $k_4 > 0$ depending only on $k_{1}$ and $k_3$.

By integration of (3.22), we find

Hence, $\mathcal{J}_{1}(t)$ blows up at most at the finite time

4.   Conclusions

Motivated by last recent mentioned papers (see ^[1,3,4]) and under some sufficient conditions, we have stated and proved the blow up in finite time of solutions for system (1.1). In the next work, we have been extend our recent work to the high dimension. Also some numerical examples have been explained in order to ensure the theory study.

Acknowledgments

The fifth author extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant (R.G.P-2/1/42).

The first, third and fourth researchers would like to thank the Deanship of Scientific Research, Qassim University for funding publication.

Conflict of interest

This work does not have any conflicts of interest.