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

: This work studies the blow up result of the solution of a coupled nonlocal singular viscoelastic equation with damping and general source terms under some suitable conditions.

(1.1) f 1 (., .),f 2 (., .): R 2 −→ R are given by with r ≥ −1, a 1 , b 1 ∈ R. D T = (0, L) × (0, T ), L, T, µ 1 and µ 2 are positive constants, g 1 , g 2 : R + → 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 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 Ω is a bounded domain of R n , (n ≥ 1) with a smooth boundary ∂Ω, p > 2, m ≥ 1.
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 (µ 1 = µ 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.

Preliminaries
We start by defining the weighted Banach space L p x = L p x ((0, L)) equipped with the norm For p = 2, we obtain the Hilbert space H = L 2 x ((0, L)) equipped the finite norm Consider also the Hilbert space V := V 1 x ((0, L)) with finite norm and As in Sobolev spaces, one can prove the following lemma 2 dx, for all v in V 0 . (2.5)
Then, there exists a small enough positive number T * such that system (1.1) admits a unique local 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 and Take a 1 = b 1 = 1 for convenience.

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.

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.

From ( 3 .k 4 >.
15) and (3.21), we finally get the required inequality J 1 (t) ≥ k 4 J 0 depending only on k 1 and k 3 .By integration of (3.22), we find Hence, J 1 (t) blows up at most at the finite time