Repeated-root dynamic-boundary heat kernel (${\mathcal{L}}_t{J}_a=e^{-x\sqrt{p}}/(\sqrt{p}-a{)}^2$)

For the dynamic boundary condition for the heat equation with $\alpha =2a$, $\beta =a^2$, the boundary resolvent denominator is $(\sqrt{p}-a{)}^2$. With the principal square root, its inverse kernel isEquivalently, $J_a={\int }_0^{\infty }r{e}^{ar}P(x+r,t)dr$ using the Heat Poisson kernel. This representation fixes the repeated-root sign and remains valid for either sign of $a$.

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.