A stability estimate for a first-order linear PDE If we have 
$$u_t + u_x =f(x,t)$$
with initial boundary conditions
$u(0,t)=0$ for $t>0$ and $u(x,0)=0$ for $0<x<R$
Can anyone tell me how to prove the stability estimate 
$$\int_0^r (u(x,t))^2 dx \leq e^t \int_0^t \int_0^R f^2(x,s)dx ds$$ where $t>0$

The method of characteristics yields
$$u(x,t)=\int_0 ^x f(s, s + t - x) d s$$ for $t\ge x$
and $$u(x,t)=\int_0 ^t f(s + x - t, s) d s$$ for $x \ge t$.
Then you can use Hölder's inequality to estimate $u^2$ in terms of an integral over $f^2$, so the integral over $u^2$ can be estimated by an integral over a quadrilateral or triangle which is a subset of $[0,R]\times[0,t]$. This shows that the inequality is true with a factor of $t$ instead of $e^t$, so that you get an even stronger estimate.
