## 5 Discussion  
The local temperature distribution \(T(t,r)\) and the local cooling rate \(-\mathcal{R}(t,r)\) determine the local structure of the glass matrix during laser-induced microstructuring. Thus, knowledge of the dynamics of the local temperature distribution \(T(t,r)\) is very important for applications associated with local laser heating, for example, for the femtosecond laser microstructuring of glasses. However, as shown in this article, the dynamics of the local temperature distribution \(T(t,r)\) significantly depends on the time dispersion of the dynamic heat capacity \(c_{dyn}(t)\) of the glass matrix. This work proposes a method for the analytical calculation of \(T(t,r)\) and \(\mathcal{R}(t,r)\) for glass-forming materials under local fast laser excitations. It is shown that the dynamics of the local temperature distribution \(T(t,r)\) caused by a laser pulse can be described by the heat equation with dynamic heat capacity \(c_{dyn}(t)\). This equation has an analytical solution for a spherically symmetric boundary value problem. Using this analytical solution, we obtained the temperature distribution \(T(t,r)\) and local cooling rate \(-\mathcal{R}(t,r)\) for the thermal parameters of borosilicate and sodium-time-silicate glasses. In fact, this analytical solution depends on the following relaxation parameters of the dynamic heat capacity \(c_{dyn}(t)\) of the material: relaxation time \(\tau_{0}\) and the distribution function \(H(\tau_{0})\), which can be obtained from broadband heat capacity spectroscopy. However, as shown in this article, the shape of the distribution function \(H(\tau_{0})\) has little effect on \(T(t,r)\) and \(\mathcal{R}(t,r)\), since with increasing \(\tau_{0}\), the influence of the time dispersion of the dynamic heat capacity \(c_{dyn}(t)\) reaches saturation. It should be noted that the influence of the time dispersion of the dynamic heat capacity is most pronounced at the beginning of the process, when \(t\) changes on a nanosecond time scale. We found that this effect is significant already at a \(\tau_{0}\) of about 0.1 us, increases with increasing \(\tau_{0}\), and reaches saturation at \(\tau_{0}\) above \({r_{0}}^{2}/D_{0}\). Thus, the influence of the time dispersion of dynamic heat capacity on the temperature distribution \(T(t,r)\) can be calculated for a fixed, sufficiently large \(\tau_{0}\) (in fact, for \(\tau_{0}\) on the order of ten or several tens of microseconds, depending on the radius of the heating zone \(r_{0}\)). As expected, the results obtained are very similar for different \(r_{0}\) values. However, as the size of the heating zone increases, the laser pulse energy \(E_{p}\) must increase in proportion to the volume \(V_{0}\) of the heating zone in order to obtain the same amplitude of the local thermal response. It has been established that the temperature distribution \(T(t,r)\) and local cooling rate \(-\mathcal{R}(t,r)\) are not affected by the heating pulse duration \(\tau_{p}\) or the shape of the heating pulse. It should be noted that the temperature perturbation \(\delta T(t)=T(t,0)-T_{in}\) in the center of the heating zone does not relax as \(\delta T\sim t^{-3/2}\), in contrast to the often-used estimate based on the fundamental solution of the Fourier heat equation. In fact, the heating pulse cannot be considered as an instantaneous point source of heat. Thus, it is necessary to integrate the thermal response over the time interval \((0,\tau_{p})\) and the volume of the heating zone, and then the resulting correct solution \(T_{FS}(t,r)\) completely coincides with the temperature distribution \(T(t,r)\) found in this article for \(\tau_{0}=0\). In addition, the solution found in this article can be used for materials with dynamic heat capacity \(c_{dyn}(t)\) at \(\tau_{0}\
eq 0\). We found that the rate of temperature change \(\mathcal{R}(t,r)\) is greatest near the periphery of the hot zone. The cooling rate reaches a maximum of \(-\mathcal{R}(t,r_{max})\) when \(r_{max}\) is slightly less than \(r_{0}\).