Blow up and global existence in a nonlinear viscoelastic wave equation

where a, b > 0, p > 2, m ≥ 1, and Ω is a bounded domain of R (n ≥ 1), with a smooth boundary ∂Ω. In the absence of the viscoelastic term (g = 0), the problem has been extensively studied and results concerning existence and nonexistence have been established. For a = 0, the source term bu |u|p−2 causes finite time blow up of solutions with negative initial energy (see [2], [8]). For b = 0, the damping term aut |ut|m−2 assures global existence for arbitrary initial data (see [7], [9]). The interaction between the damping and the source terms was first considered by Levine [10], [11] in the linear damping case (m = 2). He showed that solutions with negative initial energy blow up in finite time. Georgiev and Todorova [6] extended Levine’s result to the nonlinear damping case (m > 2). In their work, the authors introduced a different method and determined suitable relations between m and p for which there is global existence or alternatively finite time blow up. More precisely: they showed that solutions with any initial data continue to exist globally “in time” if m ≥ p and blow up in finite time if p > m and the initial energy is sufficiently negative. Without imposing the condition that the initial energy is sufficiently negative, Messaoudi [17] extended the blow up result of [6] to solutions with negative initial energy only. For results of the same nature, we refer the reader to Levine and Serrin [12], Levine, Park, and Serrin [13], and Vitillaro [19]. In the presence of the viscoelastic term (g = 0), Cavalcanti et al. [4] studied (1.1) for m = 2, and a localized damping a(x)ut (a(x) can be null on a part of the boundary). They obtained an exponential rate of decay by assuming that the kernel g is of exponential decay. This work extended the result of Zuazua [20] in which he considered (1.1) with g = 0 and the linear damping is localized. When the damping is caused only by the memory