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Consider the Cauchy problem, $u_{t} + a(x)u_{x} = 0$, $u(x, 0) = f(x)$ for $x \in \mathbb{R}$. What is an example of a smooth unbounded $a(x)$ such that the solution of this Cauchy problem is not unique?

This seems like something to solve with method of characteristics. If $a(x) := 1/x$, then I can show that the solution is not unique, but $1/x$ is not smooth on all of $\mathbb{R}$. Does anyone have any suggestions of what other functions to try?

user187437
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When parametrized by $t$, the characteristic curves satisfy the ODE $x'(t) = a(x(t))$. So, we need $a$ such that all these curves go away after some time. A convenient choice is $a(x)=x^2+1$, because the solutions of $x'(t)=x(t)^2+1$ are $x(t)=\tan(t+C)$. Whatever the initial point is, the characteristic curve does not propagate beyond $t=\pi$. Hence, the values of $u(x,t)$ for $t>\pi$ are not determined by the values $u(x,0)$.