Well defined extinction time of solutions for a class of weak-viscoelastic parabolic equation with positive initial energy

In the present paper, we consider an important problem from the point of view of application in sciences and mechanic, namely, a class of p(x)-Laplacian type parabolic equation with weak-viscoelasticity. Here, we are concerned with global in time non-existence under suitable conditions on the exponents q(x) and p(x) with positive initial energy. We show that the weak-memory term is unable to stabilize problem (1.2) under conditions (1.5) and (1.7). Our main interest in this paper arose in the first in of a to