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transport equation:

$$ \frac{d}{dt}u +cu = 0\qquad \mbox{ in } \mathbb{R}^n\times (0,\infty)$$

$$ u(x,0) = 0 \qquad \mbox{ on }\mathbb{R}^n\times \{t=0\} $$


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hint Consider the integral curve $\gamma: (0,\infty)\to\mathbb{R}^n$ given by $$\gamma(s) = (sb_1,sb_2,\ldots, sb_n)$$


$$\frac{d}{ds} u(s,\gamma(s)) = \left(\frac{\partial}{\partial t}u + b\cdot\nabla u\right)(s,\gamma(s))$$

and your equation becomes the ordinary differential equation along the curves $\gamma$ given by $\frac{d}{ds} u(s,\gamma(s)) = - cu$.

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@user7224: pretty much: you will have $u(s,\gamma(s)) = u(0,\gamma(0)) \exp (-cs)$. So now you just need to match $u(0,\gamma(0))$ to your initial data. – Willie Wong Feb 18 '11 at 17:08
For your most recent edit: if the initial data given is identically vanishing, then the only solution to the transport equation is that $u$ vanishes everywhere. – Willie Wong Feb 20 '11 at 18:53

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