Given $$g=g(x):\mathbb{R}\to \mathbb{R}$$ $g\in C^1(\mathbb{R})$ with compact support, Is It always possibile to construct $v=v(t,x) \in C^1([0,+\infty)\times\mathbb{R})$ such that $v$ has compact support and $$v(0,x)=g(x)$$ for all $x \in \mathbb{R}$?
Thanks a lot in Advance!