Time-fractional equations with reaction terms: fundamental solutions and asymptotics

We analyze the fundamental solution of a time-fractional problem, establishing existence and uniqueness in an appropriate functional space. We also focus on the one-dimensional spatial setting in the case in which the time-fractional exponent is equal to, or larger than, $\frac{1}{2}$. In this situation, we prove that the speed of invasion of the fundamental solution is at least `almost of square root type’, namely it is larger than~$c{t}^{\beta }$ for any given~$c>0$ and~$\beta \in (0,\frac{1}{2})$.