Weakened well-posedness of a hyperbolic characteristic boundary value problem

We deal with a hyperbolic characteristic boundary value problem for a Friedrichs symmetrizable system of first order with constant coefficients satisfying a weakened version of the so-called Uniform Kreiss–Lopatinskii (UKL) Condition on the boundary. The boundary value problem is weakly $L^{2}$ well-posed in the sense that it admits a unique solution satisfying an energy estimate where the failure of the UKL Condition yields a loss of regularity with respect to the data. The proof consists in splitting the original problem into two boundary value problems: a boundary value problem with a strictly dissipative boundary condition and another boundary value problem with a null source term in the interior equations. The $L^{2}$ solvability has been obtained thanks to a Fourier–Laplace analysis involving the weakened condition on the boundary.

We extend the analysis to an initial boundary value problem on a finite time interval $[0,T]$ by incorporating an arbitrary initial data. Assuming that the UKL Condition holds, we state a $L^{2}$ well-posedness result in the characteristic case.