Coercive Hamilton–Jacobi equations in domains: the twin blow-ups method

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. In this work, two blow-ups are performed simultaneously, one for each variable of the doubling variable method. We show a key one-sided Lipschitz estimate satisfied by a combination of the two blow-up limits. Both blow-up limits are a priori allowed to be infinite separately. 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.