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EXISTENCE OF SOLUTION TO CAUCHY PROBLEM OF AN EQUATION WITH DOUBLE CHARACTERISTICS

X. J. Zheng

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Abstract

This paper deals with the existence of the solution to the Cauchy problem for the equation. Respectively we have given the necessary and sufficient conditions for the Cauchy problem, which has a C~∞ solution in the cases of P≠1, 3, 5,…and P=1, 3, 5…. Furthermore, in the case of P=1, 3, 5…, some additional data in the double characteristic points x=0 for Eq. Lpu=0 have been given, and for this new problem we have also found the necessary and sufficient condition of the existence and uniqueness of the C~∞ solution.

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What this paper is about

This paper deals with the existence of the solution to the Cauchy problem for the equation. Respectively we have given the necessary and sufficient conditions for the Cauchy problem, which has a C~∞ solution in the cases of P≠1, 3, 5,…and P=1, 3, 5…. Furthermore, in the case of P=1, 3, 5…, some additional data in the double characteristic points x=0 for Eq. Lpu=0 have been given, and for this new problem we have also found the necessary and sufficient condition of the existence and uniqueness of the C~∞ solution.

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Available abstract

This paper deals with the existence of the solution to the Cauchy problem for the equation. Respectively we have given the necessary and sufficient conditions for the Cauchy problem, which has a C~∞ solution in the cases of P≠1, 3, 5,…and P=1, 3, 5…. Furthermore, in the case of P=1, 3, 5…, some additional data in the double characteristic points x=0 for Eq. Lpu=0 have been given, and for this new problem we have also found the necessary and sufficient condition of the existence and uniqueness of the C~∞ solution.

Key concepts: Uniqueness, Initial value problem, Mathematics, Cauchy problem, Cauchy boundary condition, Cauchy's convergence test, Mathematical analysis, Cauchy distribution

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