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A (classical) solution of the wave equation $$ u_{tt}-c^2u_{xx}=0,\qquad (x,t)\in\mathbb{R}\times\mathbb{R}^*_+, $$ is required to be of class $C^2$. Why?

I mean, why would one impose that all second partial derivatives, even $u_{xt}$ , which does not appear in the PDE, must be continuous?!

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1 Answer 1

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You are right, it is some overkill. I guess you could require something like $$ u\big(x, \cdot\big) \in C^2\big(\mathbb{R}\big) \land u\big(\cdot, t \big) \in C^2\big(\mathbb{R}_+\big),$$ which is a minimal less restricted case although you probably just exclude some very specially tailored counterexamples.

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    $\begingroup$ Are you using Taylor series to tailor these counterexamples? 😀 $\endgroup$ Jan 21, 2022 at 18:27
  • $\begingroup$ Forgive my English :D $\endgroup$
    – Dan Doe
    Jan 21, 2022 at 21:16

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