Blow up at well defined time for a coupled system of one spatial variable Emden-Fowler type in viscoelasticities with strong nonlinear sources

1.   Introduction

Let $x \in(r_1, r_2), t \in [1, T), T > 0, u_1 = u_1(t, x)$ and $u_2 = u_2(t, x)$. We consider a new kind of coupled system of Emden-Fowler type wave equations in viscoelasticities with strong nonlinear terms

where $i = 1, 2$ and

for $\rho > -1, r_i$ are real numbers and the scalar functions $\mu_i$ (so-called relaxation kernels) are assumed only to be nonincreasing $\mu_i \in C^1(\mathbb{R}^+, \mathbb{R}^+)$ satisfying

Many issues in physics and engineering pose problems that deal with coupled evolution equations. For example, in diffusion theory and some mechanical applications, such evolution equations are in the form of a system of nonlinear hyperbolic equations. An important example of such systems goes back to ^[13], which introduced a three-dimensional system of space similar to our system, without dissipations. (see ^{[1,11,12,16,17]}).

The Emden-Fowler equation has an impact on many astrophysics evolution phenomena. It has been poorly studied by scientists until now, essentially of the qualitative point of view.

In 1862, Land ^[15] proposed the well known Lane-Emden equation

where $p = 1.5$ and $2.5$. When $p = 1$, Eq (1.4) has a solution $u = \sin t/t$, and when $p = 5$, the explicit solution is given by $u = 1/\sqrt{1+t^2/3}$ (^[2,3,14]).

The generalization of such equation is given by

It was considered by Fowler ^[4,5,6,7] in a series of four papers during 1914–1931.

Next, the generalized Emden-Fowler equation

was studied by Atkinson et al.

Recently, M. R. Li in ^[8] considered and studied the blow-up phenomena of solutions to the Emden-Fowler type semilinear wave equation

The present research aims to extend the study of Emden-Fowler type wave equation to the case when the viscoelastic term is injected in domain $[r_1, r_2]$ where there is no result about this topic as equations and as coupled systems. Thus, a wider class of phenomena can be modeled.

The main results here are to exhibit the role of viscoelasticities, which makes our system (1.1) dissipative. When the energy is negative, the blow up of solutions in $L^2$ at finite time given by

will be the main result in Theorem 3.1.

The plan of this article is as follows. We present some notations and assumptions needed for our results in section 2. Section 3 is devoted to the blow up result of solutions.

2.   Preliminaries and position of problem

Under some suitable transformations, we can get the local existence of solutions to Eq (1.1). Taking the first transform

for $i = 1, 2$, we have

Then problem (1.1) takes the form

To make the second transformation, let

Since we have

then Eq in (2.1) can be rewritten as

and converted to

with the corresponding initial and boundary conditions. Throughout this paper, we shall write $w_{i} = w_i(t, x)$ where no confusion occurs. The following technical Lemma will play an important role.

Lemma 2.1. For any $y \in C^{1}\left(0, \ln T; H_{0}^{1}(r_1, r_2)\right)$ and $i = 1, 2$, we have

Proof. It's not hard to see

Consequently, we obtain

which implies

This completes the proof.

The modified energy associated to problem (2.2) is introduced as

and

For this energy, Lemma 2.3 leads to

As in ^[10], one can easily verify that

and

Next, we introduce the Dirichlet-Poincaré's inequality in one spatial variable.

Lemma 2.2. For any $v \in H_0^{1}(r_1, r_2)$, we have

Proof. From (2.1), we have $w(t, r_1) = w(t, r_2) = 0$. By the Fundamental Theorem of Calculus

Therefore

Recall the Cauchy-Schwar's inequality

Apply this with $f = 1, g = |w_x|$ to get

Squaring both sides gives

and finally we integrate over $[r_1, r_2]$ to give the required.

Now, in order to deal with nonlinear terms (1.2) which are considered as a sources of dissipativity in (2.2), we need the next important Lemma. This Lemma shows that the energy functional is decreasing.

Lemma 2.3. Suppose that $v\in C^{1}(0, \ln T; H_0^{1}(r_1, r_2)) \cap C^2(0, \ln T; L^2(r_1, r_2))$ is a solution of the semi-linear wave equation (2.2). Then for $t \geq 0$, we have

Proof. Taking the $L^2$ product of $(2.2)_i$ with $\partial_{t}w_{i}$ yields

for $i = 1, 2$. Adding each other, we have

Thus, by Lemma 2.1 and (2.4), we obtain

Dropping the positive terms from the left-hand side, we have

which gives the conclusion by integrating both sides with respect to $t$.

Remark 2.4. Concerning the local existence, we can follow the steps of results in ^[9] as equations, with some modifications imposed by the existence of the memories terms, where we replace the operator $\partial_{tt}-\Delta$ by $\partial_{tt}-\Delta \Big(1-\int_{0}^{t}\mu_i (t-s) \, ds \Big)$ with some conditions on the exponent $\rho$. The local existence results for one equation still valid for a coupled system in the same type.

3.   Blow up result

We prove that $(u_1, u_2)$ blows up in $L^2$ at finite time $T^{\ast}$ in the following Theorem.

Theorem 3.1. Let $\rho > -1$ and $\left(r_{2} - r_{1} \right)^2 < 8$. Suppose that

is a weak solution of (1.1) with

Assume that $l$ satisfies

Then there exists $T^{\ast}$ such that

where

Remark 3.2. Let

Now that $(r_{2} - r_{1})^2 < 8$ holds, we have

Thus by the monotonicity and $\rho > -1$, we obtain

Hence the assumptions of Theorem make sense.

Remark 3.3. In the case of $r_{1} = 0$ and $r_{2} = 1$, we introduce the example of $\mu_{i}(t)$ satisfying (1.3) and assumptions of Theorem 3.1. Let

Then we have

The condition $g(\rho) \leq l$ is equivalent to

We need to state and prove the next intermediate Lemma.

Lemma 3.4. Under the assumptions in Theorem 3.1, we have

Proof. Let

We have

by (2.3). Since

holds by (2.7), we have

by (1.3) and Lemma 2.3.

We are now ready to prove Theorem 3.1

Proof. (of Theorem 3.1)

Let

Then we have

and

By Lemma 2.4 and similar computation to Lemma 2.1 with Young's inequality

for $a, b \geq 0$ and $\theta > 0$, we have

where $\theta$ is a positive constant to be chosen later. This estimate implies that

by using Lemma 3.4. Hence we choose $\theta = 2 (\rho+2)$ to obtain

by Lemma 2.2, we have

where $g(\rho)$ is defined in Remark 3.2. Now under the assumption $e(0) > 0$, (3.1) yields

and

Here, thanks to $e(0) > 0$, $A(0) > 0$ follows. Hence we have just showed that $A(t)$ blows up. To complete the proof, we'll prove that the blow-up time $T_{1}^{\ast}$ is finite. As in ^[8], let us now set

We have only to show that $J(t)$ reaches $0$ in finite time. Then we have

and

by using Cauchy-Schwarz and Hölder's inequalities. Integrating (3.2) twice, we have

Noting that $J'(0) < 0$, we take

so that $J(t) \to 0$ as $t \to T_{1}^{\ast}$. Thus for a solution $w_{i}$ of (2.2), we obtain

as $t \to T_{1}^{\ast}$. Since

holds for all $t \in [1, \exp{T_{1}}^{\ast})$ by denoting $w_{i} = w_{i}(\tau, x)$, the conclusion follows right away together with $T^{\ast} = \exp{T_{1}^{\ast}}$.

Acknowledgement

The authors would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions to improve the paper.

Conflict of interest

The authors agree with the contents of the manuscript, and there is no conflict of interest among the authors.