In this paper, we study the graphical mean curvature flow in a warped product \(_r G/K \times I\) , where G/K is a symmetric space of compact type, I is an open interval, and r is a smooth positive function on I. If the initial hypersurface is K-equivariant, then the K-equivariance is preserved along the mean curvature flow. Here, we note that isotropy group K acts naturally on both G/K and \(_r G/K \times I\) . If the flow is graphical, then it follows from the K-equivariance of the flow that it can be described by using K-invariant functions on G/K. We derive the flow equation which these functions satisfy. By using the flow equation, we prove that the mean curvature flow exists for infinite time under the conditions that G/K is a rank one symmetric space of compact type and the warping function r satisfies certain additional properties. The proof is carried out by estimating the gradient of the K-invariant functions satisfying the flow equation.
