Abstract
Under some given conditions, we prove the explosion result of the solution of the system of nonlocal singular viscoelastic with damping and source terms on general case. This current study is a general case of the previous work of Boulaaras.
1. Introduction
During the last decades, many nonlocal problems of deterministic and parabolic partial differential equations have been studied. These equations and their systems represent the modeling of many physical phenomena related to time. These constraints can be data measured directly at the boundary or give integral boundary conditions (for instance, see [1–25]).
In this work, we investigate the blow-up of the following system of nonlinear damping term: wheregiven by with , (we get ), , , , and are given functions which will be specified later. The motivation of our work is because of some results regarding the following research paper: in [12], under some conditions suitable for the relaxation function, the author explained that solutions with initial negative energy explode in a finite time if and continue to find if , for the following studied problem:
In [4], the author studied a model describing the movement of a flexible two-dimensional viscous body on the unit disk (i.e., radial solutions) and by using some density arguments and some prior estimates, the authors demonstrated the existence and uniqueness of a generalized solution to the following problem: where and is the right-hand side that satisfied the Lipschitzian condition. Recently, in [3], the authors demonstrated the decay result of energy for a small enough initial data together with the explosion result of large initial data of the following singular problem:
That is, they obtained the blow-up properties of local solution by Georgiev-Todorova method with nonpositive initial energy. More work followed up on similar nonlocal singular viscoelastic equations and systems in [8, 9].
In this work, we continue the study on system (1). According to some given conditions, we prove the explosion result of the solution of the system of nonlocal singular viscoelastic with damping and source terms on general case, where we begin by giving basic definitions and theories about the function spaces we need, and then, we mention the theorem of local existence. Finally, we announce and prove the main result of our studied problem in (1).
1.1. Preliminaries
In this section, we introduce some functional spaces and give some lemma’s need for the remaining of this paper. Let be the weighed Banach space equipped with the norm
is, in particular, the Hilbert space of square integral functions having the finite norm
is the Hilbert space equipped with the norm
Lemma 1 (Poincare-type inequality). For any, where is some positive constant.
Remark 2. It is clear thatdefines an equivalent norm on
Lemma 3. For anyand, we havewhere is a constant depending on and only. For the and functions, assumptions are as follows:(G1): are two differentiable and nonincreasing functions with (G2): For all (G3):
Theorem 4. Suppose that (G1), (G2), and (G3) hold. Then, for alland all, problem ((1)) admits a unique local solution (u, v):for small enough.
Lemma 5. Assume that (G1), (G2), and (G3) hold and (u, v) is a solution of problem((1)); then, the energy functionalwhere
Remark 6. Multiplying the first equation in((1))byand the second equation in((1))byintegrating over, we obtain the following equation:
The definition of the norm is as follows:
From here,
Thus,
Lemma 7. There existandpositive constants such that
Lemma 8. If,
2. Blow-Up of Solution
In this section, we shall deal with the blow-up behavior of solutions for problem (1). We derive the blow-up properties of solutions of problem (1) with nonpositive initial energy by the method given in [1].
Theorem 9. Assume that (G1), (G2), and (G3) hold.and
Then, the solution of problem (1) blows up in finite time.
Proof. Since , We define ; then, ☐
We obviously substitute in (26); then,
Thus,
Equation (29) will then be used as an important data for proof of the theorem. Now, we define for small enough and
By differentiating (30), using (1) and , we obtain
By using Young inequality and from , we obtain where
From (34), and for ,
To estimate the last term in (36), we apply the three-parameter Young inequality: ,, . We take in this case:
Similarly
Substituting (38), (40), and (41) into (36), by organizing, we obtain
Since integration in estimate (40) and (41) is performed over the space, the parameter and can be a function of time; we get them as follows: where and are sufficiently large constants to be specified further. By using (43) and (44) in (42), we have
To estimate the last two terms in (45), we use (29); then,
On the other hand, since from ,
By using we can estimate the following:
Consequently, we have
Similarly
By using (51) and (52), for; we have
From (31),
From here,
Thus, by applying (23), we obtain
Substituting these inequalities in (54) and (55), in this case,
With the combination of (59) and (60), we obtain
Finally, and by considering (61), thus by organizing (45), we have which introduce the constant
Taking sufficiently large and for the positive constant , we simplify (63)
For fixed , , and , we choose so small that the following inequality holds:
Moreover, we assume that the initial data satisfy the estimate
Then, from (65), we obtain the following inequality:
On the other hand, in Equation (30), we take theof each side
Twice by applying the following inequality to (69) we have where . Now, to estimate the last two terms in (71), we, respectively, apply Holder inequality, , and Young inequality; thus,
Similarly, where . In these inequalities by collecting side by side, we obtain
We choose to get then
By applying (23), we can write
From here, we obtain Thus, by considering (78) and the following in (71), we obtain
Finally, by combining (68) and (80), we obtain the following ordinary differential inequality: obviously, where is a constant depending only , , and . This differential inequality integration over gives where we choose
Hence,
3. Conclusions
The purpose of this paper is to study the explosion result of the solution of the system of nonlocal singular viscoelastic with damping and source terms on general case. This current study is a general case of the previous work of Boulaaras in ([5]). In the next work, we will try to obtain the same result for the two-dimensional problem that allows a reasonable description of the phenomenon occurring in a three-dimensional domain. Then, we will try to prove uniqueness results of the weak solution.
Data Availability
No data were used to support the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.