- #1

evinda

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We have the Cauchy problem of the equation

$u_t+xu_x=xu, x \in \mathbb{R}, 0<t<\infty$

with some given smooth ($C^1$) function $g$ as initial value.

I want to check if the problem is well defined for each time. We know that a problem is well defined if the solution exists, is unique and depends continuously on the data of the problem.

I have computed that the solution of the problem is $u(x,t)=g(xe^{-t}) e^{x(1-e^{-t})}$.

So we have that the problem is well-defined if $g$ does not take two different values for some specific $t$, right?

But is this implied from the fact that $g$ is smooth? (Thinking)