Repeated-root unstable heat boundary mode
= Repeated-root unstable heat boundary mode
{title2=$e^{-ax+a^2t}$}
For $a>0$, the <heat equation> on a half-line with $u_t+2au_x+a^2u=0$ at the left endpoint admits $u=e^{-ax+a^2t}$. The solution is spatially decaying and temporally growing. The boundary resolvent has a double pole at $p=a^2$, allowing generalized contributions proportional to $t e^{a^2t}$. For $a<0$ the root $\sqrt p=a$ is absent on the <principal square root> branch.