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where Ω is a bounded domain in Rd (d ≥ 1) with a smooth boundary ∂Ω, α is a positive constant and g(t) is a positive function that represents the kernel of the memory term, which will be specified in Section 2. Here, we understand ∆2u to be the dispersive term. In the absence of the viscoelastic term and the dispersive term (that is, if g = α = 0), the model (1) reduces to the weakly damped wave equation utt − ∆u + ut = |u|p−1u, x ∈ Ω, t > 0. (2) The interaction between the weak damping term and the source term are considered by many authors. We refer the reader to, Haraux and Zuazua [1], Ikehata [2] and Levine [3,4]. If α = 0 and g is not trivial on R, but replacing the fourth order memory term in (1) by a weaker memory of the form ∫ t 0 g(t− τ)∆u(τ)dτ, then (1) can be rewritten as follows

Global solutions and general decay for the dispersive wave equation with memory and source terms