Integrodifferential equation which interpolates the heat equation and the wave equation (Q1813927)

The paper is concerned with the equation \[ u(t,x)=\varphi (x)+(1/ \Gamma (\alpha)) \int^ t_ 0(t-s)^{\alpha-1} \Delta u(s,x) ds \] with \(1<\alpha<2\). It is shown that the fundamental solution takes its maximum at \(x=\pm c_ \alpha t^{\alpha/2}\) for each \(t>0\) (so these points propagate with finite speed as in the wave equation case \(\alpha=2)\) and that the support of \(u(t,\cdot)\) is not compact for each \(t>0\) even when \(\varphi\) has compact support (so in that sense there is infinite speed of propagation as in the heat equation case \(\alpha=1)\).