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

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