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

Abstract

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 $c_{\sigma }\left(\left[-\epsilon ,\epsilon \right]\right)$ when $\epsilon \to 0$.