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چکیده
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This paper is concerned with the study of the nonlinear viscoelastic evolution equation with strong damping and source terms, described by \[u_{tt} - \Delta_{\mathbb{B}}u + \int_{0}^{t}g(t-\tau)\Delta_{\mathbb{B}}u(\tau)d\tau + f(x)u_{t}|u_{t}|^{m-2} = h(x)|u|^{p-2}u , \hspace{1 cm} x\in int~\mathbb{B}, t > 0,\] where $\mathbb{B}$ is a stretched manifold. First, we prove the solutions of the problem {1.1} in the cone Sobolev space $\mathcal{H}^{1,\frac{n}{2}}_{2,0}(\mathbb{B}),$ which admit a blow up in finite time for $p > m$ and positive initial energy. Then, we construct a lower bound for obtaining blow up time under appropriate assumptions on data.
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