DearMHFmembers,

consider the PDE IVP

$\displaystyle \dfrac{\partial\Phi}{\partial x}+\dfrac{\partial\Phi}{\partial t}=0$ for $\displaystyle (x,t)\in\Omega:=\{(x,t)\in\mathbb{R}^{2}:\ x\geq t\geq0\}$

under the initial condition

$\displaystyle \Phi(x,0)=\Phi_{0}(x)$ for $\displaystyle x\geq0$, where $\displaystyle \Phi_{0}\in\mathrm{C}^{1}$ is a function of exponential order,

i.e., there exists $\displaystyle M,a>0$ such that $\displaystyle |\Phi_{0}(x)|\leq M\mathrm{e}^{ax}$ for all $\displaystyle x\geq0$.

Prove that for each fixed $\displaystyle x\geq0$, the solution $\displaystyle \Phi(x,\cdot)$ is of exponential order, i.e.,

there exists $\displaystyle N_{x},b_{x}>0$ such that $\displaystyle |\Phi(x,t)|\leq N_{x}\mathrm{e}^{b_{x}t}$ for all $\displaystyle t\geq0$.

Note. Donotuse the solution $\displaystyle \Phi(x,t)=\Phi_{0}(x-t)$ directly to show it.