Dear MHF residents,

I wonder to know whether or not there is a way to prove without solving that

the solution of the initial value problem

$\displaystyle

\begin{cases}

\dfrac{\partial}{\partial t}u(t,s)+\dfrac{\partial}{\partial s}u(t,s)=0,\ (t,s)\in D:=\{(t,s):\ t\geq s\geq0\}\\

u(t,0)=u_{0}(t),\ t\in\mathbb{R}_{0}^{+},

\end{cases}

$

where $\displaystyle u_{0}$ is a function of exponential order.