Uniform decay rate estimates for a class of quasi-linear hyperbolic equations with non-linear damping and source terms

We consider the quasi-linear wave equation with non-linear damping and source terms where a, b, α, m, p ≥ 0. In this article, by using the multiplier methods, we give the precise uniform estimation of the decay rate, when the initial data is in the potential well. More precisely, when m = 0, the energy of solution decays exponentially and when m > 0, the energy of solution decays polynomially.