Math 6410 • ODE

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