Exponential decay of Heat equation solution I'm refereeing a paper and the authors go to great lengths to prove the following fact.
Let $W(t,x)$ be the solution to the linear heat equation on the half-line: $\partial_t W = D \partial_{xx} W $, $t>0$, $x>0$ with initial condition $W(0,x) = F(x)$. $D$ is a positive constant. And $F \in C^\infty$ is positive with $\int_0^\infty F(x) dx >0$.
Let $u(t,x) = e^{a t} e^{bx} W(t,x)$ where $a,b$ are constants, $b>0$. Their "new" result is:
Theorem: If $a >0$ then $u(t,x) \to \infty$ as $t \to \infty$ on compact subsets of $(0,\infty)$.
This seems standard. However, I'm not a PDE expert and haven't been able to pinpoint a reference I can include in my referee report. Can anyone help?
 A: The factor $e^{bx}$ seems to be irrelevant for the theorem. 
I haven't a reference, but it's straightforward to get an estimate, using an explicit 
formula for solution via Green's function:
$$
W(t,x)=\int_0^\infty G(x,y,t)F(y)\,dy.
$$
It can be assumed that $D=1$. For Dirichlet condition, say, 
$$
G(x,y,t)=\Gamma(x-y,t)-\Gamma(x+y,t),
$$
where 
$$
\Gamma(x,y,t)=\frac{e^{-\frac{x^2}{4 t}}}{\sqrt{4 \pi t}}, \quad t>0,
$$ 
is a fundamental solution for the heat equation.
Let $\alpha,\beta$ and $\varepsilon>0$ be s.t. $F(x)\ge \varepsilon$ on $[\alpha,\beta]$.
Since $G(x,y,t)$ is positive for $x,y,t>0$,
$$
W(t,x)\ge \int_\alpha^\beta G(x,y,t)F(y)\,dy\ge
\varepsilon\int_\alpha^\beta (\Gamma(x-y,t)-\Gamma(x+y,t))\,dx=
$$
$$
\frac\varepsilon2 \left(\text{erf}\left(\frac{x-\alpha }{2 \sqrt{t}}\right)+\text{erf}\left(\frac{\alpha +x}{2
   \sqrt{t}}\right)-\text{erf}\left(\frac{x-\beta }{2 \sqrt{t}}\right)-\text{erf}\left(\frac{\beta +x}{2
   \sqrt{t}}\right)\right)=
$$
$$
\frac{\varepsilon   x \left(\beta^2-\alpha^2\right)}{4 \sqrt{\pi }t^{3/2}}+O\left(\left(\frac{1}{t}\right)^{5/2}\right),\quad t\to+\infty,
$$
where $O$ is uniform on compact subsets of $(0,\infty)$. So the solution decreases no faster than $t^{-3/2}$ and it's enough to multiply it on power function $t^a$, $a>3/2$, to obtain growth when $t\to+\infty$.
