Nonlinear Analysis: Real World Applications 20 (2014) 14–20 Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa Blow-up result in a Cauchy viscoelastic problem with strong damping and dispersive Mohammad Kafini ∗ , Muhammad I. Mustafa Department of Mathematics and Statistics, KFUPM, Dhahran 31261, Saudi Arabia article info Article history: Received 2 January 2014 Received in revised form 26 April 2014 Accepted 28 April 2014 Keywords: Blow up Cauchy problem Finite time Relaxation function Viscoelastic abstract In this paper we consider a Cauchy problem for a nonlinear viscoelastic equation with strong damping and dispersive terms. Under certain conditions on the initial data and the relaxation function, we prove a finite-time blow-up result. © 2014 Elsevier Ltd. All rights reserved. 1. Introduction In [1], Messaoudi considered the following initial–boundary value problem:      u tt − u +  t 0 g (t − τ)u(τ)dτ + u t |u t | m−1 = u|u| p−1 , in Ω × (0, ∞) u(x, t ) = 0, x ∈ ∂ Ω, t ≥ 0 u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x ∈ Ω (1.1) where Ω is a bounded domain of R n (n ≥ 1) with a smooth boundary ∂ Ω, p > 1, m ≥ 0, and g : R + −→ R + is a positive nonincreasing function. He showed, under suitable conditions on g , that solutions with negative initial energy blow up in finite time if p > m and continue to exist if m ≥ p. This result has been later pushed, by the same author [2], to certain solutions with positive initial energy. A similar result has been also obtained by Wu [3] using a different method. In the absence of the viscoelastic term (g = 0), the problem has been extensively studied and many results concerning global existence and nonexistence have been proved. For instance, for the equation u tt − u + au t |u t | m−1 = b|u| p−1 u, in Ω × (0, ∞) (1.2) m, p ≥ 1, it is well known that, for a = 0, the source term bu|u| p−1 ,(p > 1) causes finite time blow up of solutions with negative initial energy (see [4]). The interaction between the damping and the source terms was first considered by Levine [5,6] in the linear damping case (m = 1). He showed that solutions with negative initial energy blow up in finite time. Georgiev and Todorova [7] extended Levine’s result to the nonlinear damping case (m > 1). In their work, the authors introduced a different method and showed that solutions with arbitrary negative energy continue to exist globally ‘in time’ ∗ Corresponding author. E-mail addresses: mkafini@kfupm.edu.sa (M. Kafini), mmustafa@kfupm.edu.sa (M.I. Mustafa). http://dx.doi.org/10.1016/j.nonrwa.2014.04.005 1468-1218/© 2014 Elsevier Ltd. All rights reserved.