Math 6410 • ODE

\textbf{Heat Equation.}
Let \(\Omega\subset\R^n\) be a bounded smooth domain, \(0<T<\infty\) be a constant,
\(c(x)\in C^\infty(\overline{\Omega})\) such that \(c(x)\le0\) for all \(x\) and
\(\psi(x,t)\in C^\infty(\overline{\Omega}\times[0,T])\). Let
\(u\in C^{2,1}(\overline{\Omega}\times[0,T])\) be a solution to
\[
\begin{cases}
  \dfrac{\partial u}{\partial t}=\Delta u+c(x)u,
    & \text{if }(x,t)\in\Omega\times[0,T],\\
  u(x,t)=\psi(x,t),
    & \text{if }(x,t)\in\Omega\times\{0\}\text{ or if }(x,t)\in\partial\Omega\times[0,T].
\end{cases}
\qquad(\heartsuit)
\]
\begin{enumerate}
\item[(a.)] Prove the Maximum Principle: If \(\psi(x,t)\ge0\) then the solution satisfies
\(u(x,t)\ge0\).
\item[(b.)] Show that such solutions of \((\heartsuit)\) are unique.
\end{enumerate}