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The Darboux problems in \(\mathbb{R}^ 3\) for a class of degenerating hyperbolic equations. (English) Zbl 0805.35080

Summary: We investigate some boundary value problems for degenerate hyperbolic equations in \(\mathbb{R}^ 3\) which are the three-dimensional analogues of the Darboux problems (or Cauchy-Goursat problems) in \(\mathbb{R}^ 2\). It is well known that the Darboux problems in the plane are well posed, while the same is not true for the corresponding problems in \(\mathbb{R}^ 3\). It turns out that two of the considered homogeneous problems have an infinite number of classical solutions. This means that for the solvability of the adjoint problems, the function on the right hand side has to be orthogonal to all the infinitely many classical solutions of the homogeneous problems. We define appropriate generalized solution and special function spaces where uniqueness and existence theorems hold for the considered problems in \(\mathbb{R}^ 3\) and show that especially M. H. Protter’s problem in \(\mathbb{R}^ 3\) has solutions with strong singularities on one part of the boundary even for smooth functions on the right hand side.

MSC:

35L80 Degenerate hyperbolic equations
35R25 Ill-posed problems for PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
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