Hadamard’s example and solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation

Abstract

Abstract In the theory of partial differential equations, an example constructed by J. Hadamard, which shows the instability of the solution of the Cauchy problem for the Laplace equation with respect to small changes in the initial data, is of great importance. Hadamard’s example served as the beginning of a systematic study of ill-posed problems in mathematical physics. On the other hand, the study of the Cauchy problem for the Laplace equation arises from problems of geophysics. At the same time, the question arises whether the Cauchy problem is correct for other elliptic equations including degenerate elliptic equations. We have constructed analogs of Hadamard’s example and established the incorrectness of the solution of the Cauchy problem for the Gellerstedt equation in two-dimensional and multidimensional cases. The condition of strong solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation in a cylindrical domain is found. The proof is based on the spectral properties of the Laplace operator and the properties of special functions.