Well-posedness of the Dirichlet problem for one class of degenerate multi-dimensional hyperbolic-parabolic equations

S. A. Aldashev

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 3, pp. 244-254.

Résumé

It has been shown by Hadamard that one of the fundamental problems of mathematical physics, the analysis of the behavior of oscillating string is an ill-posed problem when the boundary-value conditions are imposed on the entire boudary of the domain. As noted by A. V. Bitsadze and A. M. Nakhushev, the Dirichlet problem is ill-posed not only for the wave equation but for hyperbolic PDEs in general. This author has earlier studied the Dirichlet problem for multi-dimensional hyperbolic PDEs, where he has shown that the well-posedness of this problem crucially depends on the height of the analyzed cylindric domain. This paper, using the method developed in the authors previous papers, shows the unique solvability (and obtains an explicit form of the classical solution) of the Dirichlet problem in the cylindric domain for one class of degenerate multi-dimensional hyperbolic-parabolic equations. We also obtain a criterion for the uniqueness of the solution.

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