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Maxwell equations or Einstein field equations imply the existence of electromagnetic waves and gravitational waves. How can these waves persist on (vacuum) space if they radiate (and loose) energy?

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It's not the waves that radiate energy but the sources of the waves. The waves are the radiated energy. They carry energy away from a source. The energy moves from place to place, and becomes more and more thinly spread as it moves away from the source.

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