= Solution
For $B_0>0$, change to logarithmic magnitude and its correctly transformed <probability density function>:
$$
x=\log(B/B_0),\qquad f(x,t)=BF(B,t),\qquad f\,dx=F\,dB.
$$
Using $\partial_B=B^{-1}\partial_x$, the radial equation becomes
$$
f_t=Df_{xx}-3Df_x.
$$
This is <diffusion> with constant drift. Set $y=x-3Dt$ and $g(y,t)=f(y+3Dt,t)$; then $g_t=Dg_{yy}$. The initial <Dirac delta> transforms to $f(x,0)=\delta(x)$, so the heat-kernel solution gives
$$
f(x,t)=\frac1{\sqrt{4\pi Dt}}\exp\left[-\frac{(x-3Dt)^2}{4Dt}\right].
$$
Dividing by $B$ yields the <lognormal magnetic-field amplification> law
$$
\boxed{F(B,t)=\frac1{B\sqrt{4\pi Dt}}\exp\left[-\frac{(\log(B/B_0)-3Dt)^2}{4Dt}\right],\qquad B>0.}
$$
Expanding the square and using $B=B_0e^x$ gives the equivalent requested form
$$
\boxed{F(B,t)=\frac{e^{-(9/20)\gamma t}}{B_0\sqrt{(4/5)\pi\gamma t}}\left(\frac B{B_0}\right)^{1/2}\exp\left[-\frac{[\log(B/B_0)]^2}{(4/5)\gamma t}\right].}
$$
The logarithm is a <Gaussian random variable> with mean $3Dt$ and variance $2Dt$, which proves normalization and spreading directly. The exponential moments of that <Gaussian distribution> give $\langle B^q\rangle=B_0^qe^{Dq(q+3)t}$, independently checking the energy growth. The limit at $t\downarrow0$ is the initial <Dirac delta> distribution, not a pointwise finite function.
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