## 3 Temperature Distribution \(T(t,r)\) in Glasses under Local Thermal Perturbations
### Comparison with the Fundamental Solution of the Classical Problem  
It is often believed that the hot focal spot should spread and cool down according to the law that \(r\sim\left(D_{0}t\right)^{1/2}\) and \(T\sim t^{-3/2}\), respectively [19; 46; 80; 81]. This assumption is based on the fundamental solution of the Fourier heat equation, which can be represented by the following function [82]:  
\[G\left(t,\overrightarrow{r}\right)=\theta(t)\exp\left(-\left|\overrightarrow{ r}\right|^{2}/4D_{0}t\right)/(4\pi D_{0}t)^{3/2}, \tag{11}\]  
where \(\theta(t)\) is the Heaviside unit step function and \(r\) is the distance from the instantaneous point heat source. However, for a non-point heat source, the approximation of \(T(t,r)\) by this function is not satisfactory. The real dependence \(T(t,r)\) differs significantly from Equation (11) [45]. As expected, the temperature perturbation \(\delta T(t)=T(t,0)-T_{in}\) in the center of the heating zone relaxes in a time of the order of \(r_{0}^{2}/D_{0}\). However, \(\delta T(t)\) remains near the maximum value \(\delta T\big{(}\tau_{p}\big{)}\) at \(t<0.1\) us (see Figure 6), and only then does \(\delta T(t)\) relax over a time period on the order of \(r_{0}^{2}/D_{0}\). In contrast to the estimate based on the fundamental solution of the Fourier heat equation, the temperature perturbation \(\delta T(t)\) relaxes approximately as \(T\sim t^{-1}\) and not as \(T\sim t^{-3/2}\); see Figure 6. In order to obtain  
Figure 5: Time dependence of \(T(t,0)\) in the center of the hot zone during (**a**) and after (**b**) the heating pulse of half-sinusoidal shape at \(\tau_{p}=10\) ps and \(300\) ps, represented by filled and open symbols, respectively, at the same parameters as in Figure 3 (\(\tau_{0}=0\), \(0.1\) μs, \(1\) μs, and \(10\) μs—squares, circles, triangles up, and triangles down, respectively). Inset (**b**) shows the pulse power as a function of time for short and long pulses, represented by dotted and solid lines, respectively. the correct temperature distribution \(T(t,r)\) using the fundamental solution, it is necessary to integrate the thermal response over the time interval \((0,\tau_{p})\) and the volume of the heating zone. In fact, the solution of the generalized Cauchy problem for the classical heat equation with the heat source function \(\Phi(t,r)\) under zero initial conditions is expressed by the classical Poisson formula (see Equation (12)), where \(G\Big{(}t,\overset{\rightarrow}{r}\Big{)}\) is the fundamental solution of the classical heat equation [82]. This fundamental solution is represented by Equation (11). Thus, for the heat source \(\Phi(t,r)\) considered above, distributed in a volume with radius \(r_{0}\) and acting on the time interval \((0,\tau_{p})\), we obtain the temperature distribution \(T_{FS}(t,r)\); see Equation (12). \[T_{FS}(t,r)=\int_{0}^{\tau_{0}}\int_{0}^{\tau_{p}}\int_{0}^{\pi}\frac{\Phi_{0}F (\tau)}{\rho c_{0}}G\Big{(}t-\tau,\overset{\rightarrow}{r}-\overset{ \rightarrow}{\xi}\Big{)}2\pi sin(\theta)\xi^{2}d\xi d\tau d\theta, \tag{12}\]  
where \({(\overset{\rightarrow}{r}-\overset{\rightarrow}{\xi})}^{2}=r^{2}+\xi^{2}-2 \tau\xi\cos(\theta)\). Integration in Equation (12) was carried out using standard mathematical engineering software Mathcad 15.0. Integration over the variables \(\xi\) and \(\theta\) was carried out directly using Mathcad software. However, to avoid singularity, time integration was replaced with a discrete sum so that the difference between \(t\) and \(\tau\) was at least 0.3 ps. In fact, a sum of 30 terms was enough to obtain good accuracy with a sum normalization factor of 0.969 required to normalize the total energy of the heating pulse. In addition, the integration over all variables can be carried out directly using Mathcad software for \(t>2\tau_{p}\), which is far from the singularity of the fundamental solution.