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May 18, 2018 at 20:57 comment added Sam Zbarsky I think the condition MathStudent wants is $u(\cdot,0)$ and $\partial_t u(\cdot,0)$ both have compact support. Then $u(\cdot,t)$ will have compact support at all times $t$.
May 15, 2018 at 16:18 vote accept A random mathematician
May 11, 2018 at 7:01 comment added Robert Israel My $u(x,0) = 0$ and $1 - c(x) = 0$ have compact support.
May 10, 2018 at 20:19 review Low quality posts
May 10, 2018 at 20:23
May 10, 2018 at 20:14 comment added A random mathematician Thank you Robert. I just edited the question. I had an additional assumption that $u(x,0)$ has compact support which I had forgot to mention.
May 10, 2018 at 20:00 history answered Robert Israel CC BY-SA 4.0