Abstract
In this paper, the problem of global existence of solutions to the following initial value problem is studied: $$\left\{ \begin{gathered} u_{tt} - au_{xxt} - p(u_x )_x - \int_0^t {\lambda (t - s)q(u_x )_x ds} = f(x,t) \hfill \\ ( - \infty 0, \hfill \\ u \left| {_{t = 0} = \varphi (x), u_t } \right|_{t = 0} = \psi (x) ( - \infty< x< + \infty ) \hfill \\ \end{gathered} \right.$$ , which comes from viscoelastic mechanics. By making use of integral estimates method, it is proved that this problem has a global solution if, in addition to certain regularity assumptions on the given functions, the following conditions are satisfied: p′(s) ≥ c1 > 0, |q′(s)| ≤ const, λ(0) < 0, λ′(0) < λ2(0)
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