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It is an interesting open problem to achieve adaptive prescribed-time control for strict-feedback systems with unknown and fast or even abrupt time-varying parameters. In this paper we present a solution with the aid of several design and analysis innovations. First, by using a spatiotemporal transformation, we convert the original system operational over finite time interval into one operational over infinite time interval, allowing for Lyapunov asymptotic design and recasting prescribed-time stabilization on finite time domain into asymptotic stabilization on infinite time domain. Second, to deal with time-varying parameters with unknown variation boundaries, we use congelation of variables method and establish three separate adaptive laws for parameter estimation (two for the unknown parameters in the feedback path and one for the unknown parameter in the input path), in doing so we utilize two tuning functions to eliminate over-parametrization. Third, to achieve asymptotic convergence for the transformed system, we make use of nonlinear damping design and non-regressor-based design to cope with time-varying perturbations, and finally, we derive the prescribed-time control scheme from the asymptotic controller via inverse temporal-scale transformation. The boundedness of all closed-loop signals and control input is proved rigorously through Lyapunov analysis, squeeze theorem, and two novel lemmas built upon the method of variation of constants. Numerical simulation verifies the effectiveness of the proposed method.