Codigo De Activacion De Labeljoy
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.Q:
Proving local well-posedness of scalar wave equation in $C^{1}(\Omega)$
Is the following proof correct?
Let $u$ be a continuous solution to the scalar wave equation
$\partial^2u/\partial t^2 -\Delta u = 0$ in $\Omega \subset \mathbb{R}^n$,
$u(x,t) = 0$ for $(x,t) \in \partial \Omega \times [0,\infty)$
and $u(x,0) = f(x)$, $\frac{\partial u}{\partial t}(x,0) = g(x)$ for $x \in \Omega$.
Then $u \in C^{1}(\Omega \times [0,\infty))$ and $\partial u/\partial t$ is continuous in $\Omega \times [0,\infty)$.
Define $\phi(x,t) = t^{n/2}f(x)$, $\psi(x,t) = t^{(n-1)/2}g(x)$, $\gamma(x,t) = \partial u/\partial t(x,t)$.
Then $\phi \in C^1(\Omega \times [0,\infty))$, $\psi \in C^1(\Omega \times [0,\infty))$, $\gamma \in C^1(\Omega \times [0,\infty))$.
The function $\gamma$ is a weak solution to the heat equation
$\partial \gamma / \partial t = \Delta \gamma$ in $\Omega \times [0,\infty)$.
Choose a sequence $\gamma_j$ as follows:
Choose $\gamma_ 0b46394aab
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