Math 6420 • PDE

Let \(b(x)\) a smooth vector field on a bounded domain \(U\) in \(\mathbb{R}^n\).
Consider a smooth scalar solution \(u\) of the equation
\[
\begin{cases}
u_t=\Delta u+\nabla\cdot(b(x)u) & \text{in } U\times(0,\infty),\\
u=0 & \text{on } \partial U\times[0,\infty),\\
u(x,0)=f(x).
\end{cases}
\]
\begin{enumerate}

\item
Show that if \(\nabla \cdot b = 0\) then for all \(t > 0\)

\[
\sup_{x\in U}u(x,t)\le \sup_U |f|.
\]



\item
Show that if \(\nabla \cdot b \ne 0\) then for all \(t > 0\)
\[
\sup_{x\in U}u(x,t)\le e^{At}\sup_U |f|
\quad\text{with } A=\sup_U \nabla\cdot b.
\]


\end{enumerate}