Singular Dirichlet boundary problems for a class of fully nonlinear parabolic equations in one dimension

In this paper, we deal with the initial value problem for a class of fully nonlinear parabolic equations with a singular Dirichlet boundary condition in one space dimension. The interior equation includes, for example, a fully nonlinear $p$-Laplace type heat equation and a $\beta$-power type curvature flow. The singular Dirichlet boundary condition depicts, for example, the asymptoticness of the ends of complete curve to parallel two lines in geometric flow of graphs. We study the dependence of the existence and non-existence of solution to the problem on the interior equation and the boundedness of the initial function.