I'm stuck with the following problem.
$u_t+au_x=b(x,t),\ x \in \mathbb{R}, t>0 \\ u(x,0)=\Phi(x), x \in \mathbb{R}$
Here, $a$ is constant, and $b,\Phi$ are smooth functions. Now we assume some error in $\Phi$ and define
$v_t+av_x=b(x,t) \\ v(x,0)=\Phi(x)+\epsilon(x).$
I shall show that:
$sup_{x\in\mathbb{R},t>0} |u(x,t)-v(x,t)|=sup_{x\in\mathbb{R}}\epsilon(x).$
Can somebody give me a hint on the approach?
thank you very much! -marie