A Heat Conduction Problem on Some Semi-infinite Regions

An infinite homogeneous $d$-dimensional medium initially is at zero temperature. A heat impulse is applied at the origin, raising the temperature there to a value greater than a constant value $u_0 > 0$. The temperature at the origin then decays, and when it reaches $u_0$, another equal-sized heat impulse is applied at time $t_1$. Subsequent equal-sized heat impulses are applied at the origin at times $t_n$, $n \geq 2$, when the temperature there has decayed to $u_0$. The waiting-time sequence $\{ t_n - t_{n-1} \}$ can be defined recursively by a difference equation and its asymptotic behavior was first proposed as a conjecture by Myshkis in $1997$. In this paper we study the same heating-time problem set on semi-infinite regions $[-L, L] \times \mathbb{R}$ and $\{(x, y) : x^2 + y^2 \leq L \} \times \mathbb{R}$ with insulated boundary condition and all actions taking place at some point $\boldsymbol{p}$ which needs not be the origin.