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In the theory of partial differential equations, an example constructed by J. Hada- mard, which shows the instability of the solution of the Cauchy problem for the Laplace equa- tion with respect to small changes in the initial data, is of great importance. In the works of A.N.Tikhonov, V.Y.Arsenin, M.M.Lavrentiev and many subsequent papers, this problem is reduced to integral equations of the first kind by solving the Newton potential, various regu- larizations of the problem under consideration are given and its conditional correctness is established. In our previous work, a criterion for the strong solvability of the Cauchy problem for the Laplace equation was found by the spectral decomposition of the Laplace operator with a deviating argument. In this report, we consider a mixed Cauchy problem for two-dimensional and multidimen- sional degenerate Gellerstedst equations. Analogs of the Hadamard’s example are constructed and incorrectness of the solution of the Cauchy problem for the Gellerstedt equation in two- dimensional and multidimensional cases is established. The condition of strong solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation in a cylindrical do- main is found. The proof is based on the spectral properties of the Laplace operator and the properties of special functions.