2
$\begingroup$

Please give me a hint or an idea on how to approach the following problem:

Consider the following PDE $ \frac{\partial f}{\partial t}+v \frac{\partial f}{\partial x}+f=\chi_{n(x,t)}(v),\ \ x,v \in \Bbb{R}$ $ f(x,v,0)=f_0(x,v)\geq 0,\ $ where $f_0 \in L^1(\Bbb{R}^2)$, $ n(x,t)=\int_\Bbb{R} f(x,v,t)dv;\ \ \ \ \chi_n(v)=\begin{cases} 1& 0\leq v\leq n(x,t) \\ 0 & \text{elseway} \end{cases}$

I am asked to prove that $\|f(\cdot,\cdot,t)\|_{L^1(\Bbb{R}^2)}=\|f_0\|_{L^1(\Bbb{R}^2)}$. Please give me a few ideas as to where to start. Thank you.

0 Answers 0