  In this note, we consider an evolution coercive Hamilton-Jacobi equation
posed in a domain and supplemented with a boundary condition. We are interested
in proving a comparison principle in the case where the time and the (normal)
gradient variables are strongly coupled at the boundary. We elaborate on a
method introduced by P.-L. Lions and P. Souganidis (Atti Accad. Naz. Lincei,
2017) to extend their comparison principle to more general boundary conditions
and to Hamiltonians that are not globally Lipschitz continuous in the time
variable. Their argument relies on a single blow-up procedure after rescaling
the semi-solutions to be compared. We refer to our technique as the twin
blow-ups method since two blow-ups are performed simultaneously, one for each
variable of the doubling variable method. The Lipschitz regularity of the
regularized subsolution provides a key Lipschitz inequality satisfied by the
two blow-up limits, that are a priori allowed to be infinite. For expository
reasons, the result is presented here in the framework of space dimension one
and the general case is treated in a companion paper.
