The definition of hyperbolic equation by a prescribed vector field is introduced for linear differential equation of the second order. The Cauchy problem with prescribed boundary conditions is considered for such equations. The theorems of existence and uniqueness of a strong solution to the given problem are proved by the method of energy inequalities and mollifiers with variable step.
Korzyuk, V. I., & Cheb, E. (2006). Cauchy problem for a linear hyperbolic equation of the second order. Mathematical Modelling and Analysis, 11(3), 275-294. https://doi.org/10.3846/13926292.2006.9637318
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