It's a promblem from Arnold ODE book.
Find the largest value of $t$ for which the solution of the Cauchy problem $$ u_{t}+u u_{x}=-\sin x,\left.\quad u\right|_{t=0}=0 $$ can be continued to $[0, t)$.
I tried to find characteristics with $$\left.\quad u\right|_{t=0}=0$$But i have a bit of trouble with it. Thanks in advance.