Let $a,h\in\mathcal{C}^1(\mathbb{R})$. I found out that a solution of the problem $u_t+a(u)u_x=0$, $u(0,x)=h(x)$ for $x\in\mathbb{R}$ can be represented (implicitly) by $$u(t,x)=h(x-ta(u(t,x))).$$ Now, I have to prove that this solution $u$ exists as a function of class $\mathcal{C}^1$ iff $a(h(s))$ is a monotonically increasing function. Does someone have an idea?
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Hint: Intermediate value theorem – Matthew Cassell Jan 19 '15 at 09:15
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@Mattos: Thanks, but to which function should it be applied? I did not get the idea - all my attempts failed so far. – JohnSmith Jan 19 '15 at 14:38
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Cf. this related post. – EditPiAf Sep 03 '18 at 16:30