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

Assume a classical solution continuous up to the relevant boundary. At an interior positive maximum, $u_t=0$ and $u_{xx}\le 0$, hence $Lu=u_t-u_{xx}+u>0$, a contradiction. At a positive maximum on the top edge $t=\tau$, away from the spatial endpoints, the left time derivative satisfies $u_t\ge 0$, while again $u_{xx}\le 0$; the same contradiction applies. Thus a positive maximum can occur only on the initial or lateral parabolic boundary.

For two solutions with the same data, their difference $w$ satisfies $Lw=0$ and vanishes on that parabolic boundary. The weak parabolic maximum principle prohibits a positive maximum of $w$ and also of $-w$. Therefore $w=0$, proving at most one solution. The positive reaction term is what gives the immediate strict contradiction at a positive maximum.

Use the Fourier transform convention $\hat{u}(k,t)={\int }_{\mathbb{R}}{e}^{-ikx}u(x,t)dx$. Then the equation and Dirac delta initial condition becomeso $\hat{u}=e^{-(k^2+1)t}$. The inverse Fourier transform givesComplete the square as $-t{k}^2+ikx=-t(k-ix/(2t){)}^2-x^2/(4t)$ and use the permitted complex-shift Gaussian integral. Thus the heat kernel with constant absorption isIts mass is $e^{-t}$, consistent with constant absorption, and it tends to the Dirac delta as a distribution as $t\downarrow 0$.

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.