= Solution
Under the usual existence and boundary hypotheses for the <Laplace transform>,
$$
\boxed{\overline{f'}(p)=p\bar f(p)-f(0+),\qquad\overline{tf}(p)=-\frac d{dp}\bar f(p).}
$$
The first formula requires a finite right limit and sufficient regularity for <integration by parts>; it is not to be applied blindly to the singular function $t^{-1/2}$. For that function and real $p>0$, substitute $s=\sqrt{pt}$:
$$
\int_0^\infty e^{-pt}t^{-1/2}dt=\frac2{\sqrt p}\int_0^\infty e^{-s^2}ds=\sqrt{\pi/p}.
$$
Use the second transform identity to obtain
$$
\boxed{\mathcal L\{\sqrt t\}(p)=\frac{\sqrt\pi}{2p^{3/2}}.}
$$
Take the time transform of the <heat equation>, with <differentiation> and boundary limits interpreted in the standard decaying heat-solution class. The initial value is zero, so $v_{xx}=p v-u(x,0)=pv$. Transforming the constant flux gives $v_x(0,p)=-1/p$. Decay selects the decreasing solution of the transformed <ordinary differential equation>, giving
$$
v(x,p)=B(p)e^{-\sqrt p\,x},\qquad-\sqrt p\,B(p)=-1/p.
$$
Consequently
$$
\boxed{v(x,p)=p^{-3/2}e^{-x\sqrt p},\qquad u(0,t)=2\sqrt{t/\pi}.}
$$
The boundary value follows by comparing $v(0,p)=p^{-3/2}$ with the transform of $\sqrt t$.
One can also check the decay and transform steps without relying on a merely pointwise limit under an <improper integral>. Using the <complementary error function>, the classical <heat equation>[heat] solution is
$$
u(x,t)=2\sqrt{t/\pi}\,e^{-x^2/(4t)}-x\,\operatorname{erfc}\left(\frac{x}{2\sqrt t}\right)=\int_x^\infty\operatorname{erfc}\left(\frac{s}{2\sqrt t}\right)ds.
$$
<Differentiation> verifies $u_{xx}=u_t$, $u_x(0,t)=-1$, and $u(x,0)=0$ for $x>0$. Its integral representation gives $0\le u(x,t)\le2\sqrt{t/\pi}$ and pointwise decay as $x\to\infty$. Thus $e^{-pt}2\sqrt{t/\pi}$ supplies an integrable majorant, proving $v(x,p)\to0$ by <dominated convergence>. The same representation gives the required regularity for the transforms. This explicitly verifies that the solution selected by the decaying transformed equation lies in the intended heat-solution class.
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