Boundary behavior of solutions to the Dirichlet problem for the heat equation in a domain whose lateral boundary satisfies the Hölder condition with exponent less than $1/2$

For the heat equation with one space variable, we examine solutions of the first boundary-value problem in a domain whose lateral boundary possesses a model singularity, namely, the curve describing the lateral boundary is smooth everywhere except for one point and belongs to the Hölder class with exponent less than $1 /2$. We prove that if a solution is positive in some neighborhood of the singular point and vanishes on the lateral boundary in this neighborhood, then the first derivative of this solution unboundedly increases in any neighbourhood of the singular point.