Math 6410 • ODE

Consider the global Cauchy problem
\[
  u_y=Du_{xx}+bu_x+cu,\quad x\in\R,\ t>0;\qquad u(0,x)=g(x),
\]
where \(b\), \(c\) \(D\) are constants such that \(D>0\).
\begin{enumerate}
\item[(a)] Convert to the standard heat equation by setting
\(v(t,x)=u(t,x)e^{hx+kt}\) for appropriately chosen \(h\) and \(k\).
\item[(b)] Solve the heat equation for \(v\) using a transform method.
\item[(c)] Hence show that if \(c<0\) and \(g\) is bounded, then
\(y(t,x)\to0\) as \(t\to\infty\).
\end{enumerate}