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Apr 21, 2022 at 7:59 vote accept Desura
Apr 15, 2022 at 19:02 comment added Willie Wong My answer below side-steps your second problem. But using a similar argument using Strichartz inequalities, you should be able to prove persistence of higher regularity: that if you have an energy space solution with data in $H^k$, then the solution is also in $H^k$. Combining this with an approximation argument your second problem should also have a positive solution.
Apr 15, 2022 at 18:53 answer added Willie Wong timeline score: 4
Apr 15, 2022 at 12:37 history asked Desura CC BY-SA 4.0