BOUNDARY VALUE PROBLEM FOR A THIRD-ORDER EQUATION WTH MULTIPLE CHARACTERISTICS

Abstrak

This article investigates the well-posedness of a boundary value problem for a class of third-order nonlinear partial differential equations characterized by multiple characteristics. Equations of this type frequently appear in hydrodynamics, plasma physics, and acoustic wave propagation problems, where higher-order dispersion and nonlinear interactions play a decisive role. The primary mathematical objectives of the study are to prove the existence and uniqueness of solutions within appropriate function spaces. An energy method is employed to establish uniqueness. By constructing a suitable energy functional and deriving refined energy estimates, it is shown that the corresponding homogeneous problem possesses only the trivial solution, thereby guaranteeing uniqueness under the prescribed boundary conditions. The method of continuation with respect to a parameter is used to prove existence. Introducing a continuous parameter—which allows the original nonlinear problem to be deformed into a simpler linear analogue with a known solution—enables the establishment of global classical solvability. This approach relies on deriving uniform a priori estimates for the family of parameterized solutions, ensuring that the set of parameters for which the problem is solvable is both open and closed within the parameter space. Furthermore, the interaction between multiple characteristics and nonlinear terms is analyzed in detail, identifying the structural conditions necessary to prevent a loss of solution smoothness. The results presented extend known fundamental existence and uniqueness theorems to a broader class of third-order nonlinear differential equations.