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Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$ is a $C^1$ function which satisfied the following differential inequality: $\frac{df}{dt}\leq C(f+f^{\frac{3}{2}}).$ If $f>0$ and $f(t)\rightarrow 0$ as $t\rightarrow\infty$, then is $f$ bounded?

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No, any monotonically decreasing positive $C^1$ function that goes to infinity for $t\to-\infty$ is a counterexample, for instance

$f(t)=\begin{cases}1-t&t\lt0\\\mathrm e^{-t}&t\ge0\;.\end{cases}$

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    @Srivatsan Narayanan: Yes, you are right! Actually I want to see if $\frac{df}{dt}$ is bounded or not as $t\rightarrow\infty$.2011-09-16