This paper considers the problem of finite-time stabilization of coupled reaction-diffusion partial differential equations by means of boundary time-varying feedbacks; moreover, the time of convergence can be prescribed in the design. The design of time-varying feedbacks is carried out based on the backstepping approach. Selecting a suitable target system with time varying-coefficients, the resulting kernel of the backstepping transformation is time-varying which provides the control feedback to be time-varying as well. The target system turns out to be fixed-time stable and two cases for the control design are pointed out in order to obtain either boundedness or fixed-time convergence of the original system. A simulation example is presented to illustrate the results.