|
Suppose I have some smooth $t$-dependent function $x(t)$ in $\mathbb{R}^n$ for which I have the property \begin{equation} \langle x, \dot{x} \rangle \; \; \leq C || x||^2 \end{equation} where $\dot{x}$ is the derivative of $x$ and $\langle \cdot, \cdot \rangle$ is the natural dot product. What kind of information does this property then give? I would say something like boundedness but I am worrying that in the kernel of $\langle \cdot, \dot{x} \rangle$ things get unbounded. Any help is welcome! |
|||
|
|
|
Start by recording that $$\frac{d}{dt}\langle x(t), x(t)\rangle=2\langle \dot{x}(t), x(t)\rangle.$$ So, writing $n(t)=\langle x(t), x(t)\rangle$, the assumption can be rewitten as follows: $$\frac{dn}{dt}(t)\le \frac{C}{2}n(t).$$ You can then apply the differential form of the Gronwall's lemma. The conclusion is that $$n(t)\le n(0)e^{\frac{C}{2}t},$$ that is $\lVert x(t)\rVert^2\le \lVert x(0)\rVert^2 e^{\frac{C}{2}t}$. So the function $x(t)$ is not necessarily bounded (for an example of this let $n=1$ and take $x(t)=e^t$) but it can have at most an exponential growth. |
||||
|
|
