= Solution
Taking the spatial <Fourier transform> of the <diffusion equation> gives $\widetilde\theta_t=-Dk^2\widetilde\theta$ and $\widetilde\theta(k,0)=\widetilde g(k)$. Hence $\widetilde\theta=e^{-Dk^2t}\widetilde g$. <Fourier inversion> and the Gaussian transform from part (a), or direct evaluation of the inverse Gaussian integral, give the <heat kernel>
$$
K_t(x)=\frac1{\sqrt{4\pi Dt}}e^{-x^2/(4Dt)}.
$$
Substituting the integral for $\widetilde g$ and interchanging absolutely integrable terms produces the <convolution> solution
$$
\boxed{\theta(x,t)=\frac1{\sqrt{4\pi Dt}}
\int_{\mathbb R}g(\xi)e^{-(x-\xi)^2/(4Dt)}\,d\xi,\qquad t>0.}
$$
For the stated nonnegative compactly supported data, understood to be nonzero on a set of positive measure, the kernel is strictly positive everywhere. Therefore $\theta(x,t)>0$ for every real $x$ and every $t>0$. \b[Property P holds: diffusion has <infinite propagation speed>.] The distant value can be extremely small, but is not zero. For continuous initial data, any nontrivial nonnegative $g$ automatically has the required positive-measure set.
Back to article page
