Explicit positive solutions to $G$-heat equations and the application to $G$-capacities

In this paper, we study a class of explicit positive solutions to $G$-heat equations by solving second order nonlinear ordinary differential equations. Based on the positive solutions, we give the sharp order of $G$-capacity $c_{\sigma}\left( \left[ -\varepsilon,\varepsilon \right] \right) $ when $\varepsilon \rightarrow0$.