# Stability in PDE given an inequality

I having troubles proving stability on PDE.

$u_t +au_x = f(x,t)$ for $0<x<R$ and $t>0$

$u(0,t)=0$ for $t>0$

$u(x,0)=0$ for $0<x<R$

I have manage to get:

$\int_0^R \! u^2(x,t) \, \mathrm{d} x \leq e^t \int_0^t \int_0^R \! f^2(x,t) \, \mathrm{d} x \mathrm{d} s$

How does that implies stability?

with $L^2 norm$

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What definition of stability are you using? (by the way, try and accept some answers to your questions. you have 0% accept rate) –  Beni Bogosel Sep 27 '12 at 8:00
stability respect initial condition ^^ :p –  Maria Sep 27 '12 at 17:04
Still much too vague. Stability means that "small" change of initial data will result in "small" change in the solution. A lot depends on how smallness is measured. If you use $L^2$ norms, then (by linearity) you are done already. –  user31373 Sep 28 '12 at 3:08