Gradient Yamabe Solitons on Warped Products

The special nature of gradient Yamabe soliton equation which was first observed by Cao-Sun-Zhang\cite{CaoSunZhang} shows that a complete gradient Yamabe soliton with non-constant potential function is either defined on the Euclidean space with rotational symmetry, or on the warped product of the real line with a manifold of constant scalar curvature. In this paper we consider the classification in the latter case. We show that a complete gradient steady Yamabe soliton on warped product is necessarily isometric to the Riemannian product. In the shrinking case, we show that there is a continuous family of complete gradient Yamabe shrinkers on warped products which are not isometric to the Riemannian product in dimension three and higher.