Blow up of solutions of two singular nonlinear viscoelastic equations with general source and localized frictional damping terms

This work studies the blow-up result of the solution of a coupled nonlocal singular viscoelastic equation with general source and localized frictional damping terms under some suitable conditions. This work is a natural continuation of the previous recent articles by Boulaaras et al. (Appl. Anal., 2020, https://doi.org/10.1080/00036811. 2020.1760250; Math. Methods Appl. Sci. 43:6140–6164, 2020; Topol. Methods Nonlinear Anal., 2020, https://doi.org/10.12775/TMNA.2020.014).


Introduction
This paper is devoted to a study of the blow-up of the following system of two singular nonlinear viscoelastic equations: where and Q = (0, L) × (0, T), L < ∞, T < ∞, μ ∈ C 1 ((0, L)), g 1 (·), g 2 (·) : R + → R + and f 1 (·, ·), f 2 (·, ·) : R 2 − → R are functions given in (2). The problems related with localized frictional damping have been extensively studied by many teams [5], where the authors obtained an exponential rate of decay for the solution of the viscoelastic nonlinear wave equation: for a damping term a(x)u t that may be null for some part of the domain.
We used the techniques of [5], and we have proved in [3] the existence of a global solution using the potential well theory for the following viscoelastic system with nonlocal boundary condition and localized frictional damping: Very recently, in [2] we have studied the following singular one-dimensional nonlinear equations that arise in generalized viscoelasticity with long-term memory: Also in the field of blow-up, in [14], the authors studied the blow-up in finite time of solutions of an initial boundary value problem with nonlocal boundary conditions for a system of nonlinear singular viscoelastic equations.
In view of the articles mentioned above in [2,3,5] and a supplement to our recent study in [2], much less effort has been devoted to the blow-up of solutions of two singular nonlinear viscoelastic equations, where nonlocal boundary conditions, general source terms and localized frictional damping are considered.
The structure of the work is as follows: 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 the main result, which under suitable conditions gives the blowup in finite time of solutions for system 1.

Preliminaries
Let L p x = L p x ((0, L)) be the weighted Banach space equipped with the norm Let H = L 2 x ((0, L)) be the Hilbert space of square integral functions having the finite norm L)) be the Hilbert space equipped with the norm and and Remark 2 It is clear that u V 0 = u x H defines an equivalent norm on V 0 .
where C * is a constant depending on L and p only.
We prove the blow-up result under the following suitable assumptions.

Theorem 4 Assume
Then, for any for T * > 0 small enough.
We take a 1 = b 1 = 1 for convenience.

Blow-up
In this section, we prove the blow-up result of solution of problem (1). Now we define the functional Theorem 8 Assume (11)- (13), and (14) hold. Assume further that E(0) < 0, then the solution of problem (1) blows up in finite time.
By integration of (38), we obtain Hence, K(t) blows up in time Then the proof is completed.

Conclusion
Mixed non-local problems for hyperbolic and parabolic PDEs have been studied intensively in recent decades. Such equations or systems with constraints modelize many timedependant physical phenomena. These constraints can be data measured directly on the boundary or giving integral boundary conditions (see for example [1,[4][5][6][7][10][11][12][13]). In view of the articles mentioned above in [2,3,5] and a supplement to our recent study in [2,8], we have proved in this work the blow-up of solutions of two singular nonlinear viscoelastic equations, where nonlocal boundary conditions, general source terms and localized frictional damping are considered.