= Solution
Assume a classical solution continuous up to the relevant boundary. At an interior positive maximum, $u_t=0$ and $u_{xx}\leq0$, 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 $u_t\geq0$, while again $u_{xx}\leq0$; 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 \b[at most one solution]. The positive reaction term is what gives the immediate strict contradiction at a positive maximum.
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