Repeated-root unstable heat boundary mode ($e^{-ax+a^{2}t}$)

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