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Weakened well-posedness of a hyperbolic characteristic boundary value problem

Tom 126 / 2021

Sihame Brahimi, Ahmed Zerrouk Mokrane Annales Polonici Mathematici 126 (2021), 197-214 MSC: Primary 35L40; Secondary 35L50. DOI: 10.4064/ap200810-26-5 Opublikowany online: 19 September 2021

Streszczenie

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.

Autorzy

  • Sihame BrahimiLTM Laboratory, Department of Mathematics
    Faculty of Mathematics and Computer Science
    University of Batna 2
    Batna, Algeria
    e-mail
  • Ahmed Zerrouk MokraneLTM Laboratory, Department of Mathematics
    Faculty of Mathematics and Computer Science
    University of Batna 2
    Batna, Algeria
    e-mail

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