Let

${\Omega}$ be a bounded domain in

${\mathbb{R}}^{N}$ and let A be a uniformly elliptic operator on

${\Omega}$. The author considers the parabolic problem

${u}_{t}-Au=0$ in

${{\Omega}}_{\infty}={\Omega}\times (0,\infty )$,

$u(x,0)={u}_{0}\left(x\right)$ for

$x\in {\Omega}$ where

${u}_{0}\neg \equiv 0$ for all

${u}_{0}$ in C(

$\overline{{\Omega}}$), and

$u(x,t)={\int}_{{\Omega}}f(x,y)u(y,t)dy$,

$0<t<\infty $, where f is a continuous function defined for

$x\in \partial {\Omega}$,

$y\in \overline{{\Omega}}$ and such that, for every

$x\in \partial {\Omega}$,

${\int}_{{\Omega}}\left|f(x,y)\right|dy\le \rho <1$. He appeals to the maximum principle to prove that the problem has a unique solution u in C(

${\overline{{\Omega}}}_{\infty})$; that

$U\left(t\right)={max}_{x\in \overline{{\Omega}}}\left|u(x,t)\right|$ is monotone decreasing in t; that there are constants C,

$\gamma $ such that for all

$t>0$, U(t)

$\le C{e}^{-\gamma t}$; and that there is a

${T}_{*}$ with

$0<{T}_{*}\le \infty $ such that U(t) is strictly decreasing for

$0<t<{T}_{*}$ whereas U(t)

$\equiv 0$ for

$t>{T}_{*}$. Imposing added conditions on the coefficients of A, on the boundary

$\partial {\Omega}$, and on an extension of the function f, the author then shows that for any

$x\in {\Omega}$, u(x,t) is analytic in t,

$0<t<\infty $, and that U(t) is strictly decreasing in t for all

$t>0$ so that U(t) does not vanish in finite time.