I have the following conservation law in my hand:
$\partial_{t} u + \partial_{x}f(u) = -u$, with the associated initial data $u(x,0) = u_{0}(x)$ - where $u_{0} \in C^{1}$.
I have to show classical solution ($u \in C^{1}$) of the equation is unique.
My thoughts
For me the obvious approach was to show the $L^{1}$-contraction, i.e. assuming $\exists$ two $C^{1}$-functions $u_{1}$ and $u_{2}$ which solve the above conservation law with initial data $u_{0,1}(x)$ and $u_{0,2}(x)$ to show
$||u_{1}(\cdot,t) - u_{2}(\cdot,t)||_{L^{1}(\mathbb{R})} \leq ||u_{0,1}(\cdot) - u_{0,2}(\cdot)||_{L^{1}(\mathbb{R})}$.
With this aim in mind I subtracted two conservation laws given by $u_{1}$ and $u_{2}$ and multiply the resulting equation by a smooth function $\eta'(w)$ where $w = u_{1} - u_{2}$ and $\eta$ is a smooth approximation of modulus.
Can someone show me how to estimate the terms $\eta'(w) \partial_{x}(f(u_{1}) - f(u_{2}))$ and $\eta'(w)w$ ??
Thanks in advance!! :)