## 1 Introduction
For example, in silicate glasses at \(t=0.5\) ns, the distance \(tV_{L}\) is about 3 \(\upmu\)m at the longitudinal speed of sound \(V_{L}\approx 6\cdot 10^{3}\) m/s [58] and \(\sqrt{tD_{0}}\approx 10\) nm at the thermal diffusion coefficient \(D_{0}\) of about 3\(\cdot 10^{-7}\) m\({}^{2}\)/s [56; 62; 64]. It Is noteworthy that an important property of glass-forming materials is the long-term relaxation of the specific heat capacity with rapid changes in temperature. Thus, the heat capacity of glass-forming materials must be considered as a function of time \(c_{dyn}(t)\)[65; 66; 67]. In fact, if a glass-forming material is heated to a liquid state with heat capacity \(c_{L}\) from a solid glassy state with heat capacity \(c_{S}\), then the heat capacity of the material does not change immediately with the change in temperature but slowly relaxes from \(c_{S}\) to \(c_{L}\); see below. The long-term relaxation of the dynamic heat capacity \(c_{dyn}(t)\) of glasses is due to the slow exchange of energy between different degrees of freedom in glasses. Thus, the thermal response of glass-forming materials to a thermal perturbation at time \(t\) depends on the temperature at earlier times. The effect of the long-term relaxation of dynamic heat capacity significantly affects the dynamics of local temperature changes \(T(t,r)\)[68; 69]. In turn, the rate of change in local temperature \(T(t,r)\) can significantly affect the microstructuring of glass-forming materials [19; 37; 43; 44; 70]. An experimental study of the dynamics of the temperature distribution \(T(t,r)\) inside a glass matrix under the action of fast laser thermal perturbations is very difficult. Therefore, it is necessary to develop theoretical models of thermal processes occurring during the laser-induced structuring of glasses. In this paper, we will focus on modeling such thermal processes, taking into account the relaxation effect of the dynamic heat capacity \(c_{dyn}(t)\). The dynamic behavior of glass-forming materials under fast local thermal perturbations can be described using the integro-differential heat equation with \"memory\" [68; 69]. This equation has an analytical solution, at least in spherical, cylindrical, and planar geometries [68; 69; 71]. This work aims to determine the dynamics of local temperature changes, especially the local rate of the temperature change \(\mathcal{R}(t,r)\) associated with laser-induced thermal excitations during laser processing of glasses. An analytical method for determining \(T(t,r)\) and \(\mathcal{R}(t,r)\) has been developed. The knowledge obtained can be useful for various technologies related to laser-induced microstructuring. In the first part of the article, the heat equation with dynamic heat capacity \(c_{dyn}(t)\) is considered. An analytical solution to this equation in general form was presented in our previous work [69], and then this method was applied to describe local thermal disturbances in supercooled glass-forming liquids and polymers during the nucleation of the crystalline phase [70]. We are currently using this method to study the rate of temperature change \(\mathcal{R}(t,r)\) inside glasses under local fast laser excitations during laser-induced microstructuring. An analysis of the dynamic heat capacity \(c_{dyn}(t)\) of glass-forming materials was conducted, and an analytical solution of the heat equation with dynamic heat capacity for a spherically symmetric problem was constructed. Then, the temperature distribution \(T(t,r)\) was calculated for glasses with dynamic heat capacity \(c_{dyn}(t)\) under local fast laser excitations. Examples with borosilicate and sodium-lime-silicate glasses are considered at different heating pulses and dynamic heat capacity parameters. Finally, the local distribution of the cooling rate \(-\mathcal{R}(t,r)\) and its influence on the microstructuring processes of glasses is discussed.