Math 6410 • ODE

\textbf{Heat Equation.}
Given the constant \(\beta\in\R\), consider the initial value problem
\[
\begin{aligned}
  u_t &= \beta\Delta u,\quad x\in\R^n,\quad t>0,\\
  u(x,0) &= g(x),\quad x\in\R^n.
\end{aligned}
\]
\begin{enumerate}
\item[(a)] Assume \(\beta>0\) and let \(g\in C(\R^n)\) be bounded. Prove that there exists a
bounded solution \(u(x,t)\) which is \(C^\infty\) for all \(t>0\). Under what condition are
solutions unique?
\item[(b)] Assume \(\beta<0\) and let \(g\in C^\infty(\R^n)\) be bounded. Are solutions
\(u(x,t)\) necessarily bounded and continuous for \(t>0\)? Prove, or provide a
counterexample.
\end{enumerate}