Math 6410 • ODE

\textbf{Heat Equation.}
Let \(\Omega\subset\R^n\) be a bounded smooth domain, and \(0<T<\infty\), \(c\in\R\) be
constants. Let \(\psi(x)\in C^\infty_0(\overline{\Omega})\). Let
\(u\in C^{2,1}(\overline{\Omega}\times[0,T])\) be a solution to
\[
\begin{cases}
  \dfrac{\partial u}{\partial t}=\Delta u+cu, & \text{if }(x,t)\in\Omega\times[0,T],\\
  u(x,0)=\psi(x), & \text{if }x\in\Omega,\\
  u(x,t)=0, & \text{if }(x,t)\in\partial\Omega\times[0,T].
\end{cases}
\tag{3.}
\]
\begin{enumerate}
\item[(a.)] Show that the solution \(u(x,t)\) is bounded on \(\overline{\Omega}\times[0,T]\)
in terms of \(c\), \(\sup_{x\in\Omega}\lvert\psi(x)\rvert\) and \(t\). [You may assume \(c\le0\)
but it is true for any \(c\).]
\item[(b.)] Show that such solutions of (3.) depends continuously on \(\psi\).
\end{enumerate}