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If every classical/strong solution to a PDE with boundary values condition is a also a weak solution, then why does Brezis bother with proving the existence of a weak solution to the heat eqution knowing that a classical one exists? (chapter 10). I'm guessing it's due to the initial value $u(x,0)$ belonging to $L^2(\Omega)$ ? while the classical formulation requires it to be continuous everywhere ?

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    $\begingroup$ Well maybe there would be no point if you knew there was a classical solution, but you don’t know that. The point is that it’s easier to prove there is a weak solution and to argue that a weak solution must actually be a classical solution (via regularity theory) than to directly prove the existence of a classical solution $\endgroup$
    – JackT
    May 15 at 13:30
  • $\begingroup$ @JackT isn't the solution obtained by separation of variables/ fourier method a classical solution ? $\endgroup$ May 15 at 15:04
  • $\begingroup$ @user19645873 Yes but such computing such "simple" solutions is not possible, in general. $\endgroup$
    – K.defaoite
    May 15 at 21:20
  • $\begingroup$ The key here is that these methods work for any $\Omega$, whereas classical techniques require specific forms of $\Omega$ to solve the resulting equations $\endgroup$
    – whpowell96
    May 17 at 1:25

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