On the global existence and asymptotic behavior of the solution of a nonlinear wave equation with past history

In this paper, we consider a wave equation with past history and a nonlinear source term. We minimize the conditions imposed on the relaxation function g by assuming that g satisfies g′(t) ≤ -ζ(t)G(g(t)), where the two functions ζ and G satisfy some conditions. Using this minimal condition on the relaxation function g, we establish the local and global existence results and general decay rate results. This assumption recovers the exponential and polynomial rates when G(s) = sp and p covers the full admissible range 1,2. Moreover, we delete some assumptions on the boundedness of initial data used in many earlier papers in the literature. In fact, our results generalize, extend, and improve many earlier results in the literature.