Past exam of the differential equations and applications course of the University of Oxford / 2011 / B / 5 / c

For a fixed observation point $(\xi ,\tau )$, use the backward adjoint Green function with $L^{\ast }=-{\partial }_t-{\partial }_x^2+1$. It satisfies $L^{\ast }G=0$ for $t<\tau$, decays with its spatial derivative as $|x|\to \infty$, and has terminal distribution $G(x,{\tau }^{-};\xi ,\tau )=\delta (x-\xi )$. Extending it by zero for $t>\tau$ gives $L^{\ast }G=\delta (x-\xi )\delta (t-\tau )$.

For $0<t<\tau$, integration by parts provesEvaluating this constant pairing at the initial and terminal times yieldsBy translating the heat kernel with constant absorption and reversing the time difference,The terminal singularity is understood as a distribution; the representation follows for smooth decaying data and extends to standard integrable data by the heat-kernel approximation argument.