Boundary Problems for a Third-Order Equations with Changing Time Direction

Boundary problems for nonclassical partial differential equations, coefficients in the main part of the sign change that occurs during many applications, particularly in physics, the description processes of diffusion and transfer, in geometry and population genetics, fluid dynamics, as well as many other areas. The work is devoted to research solvability of boundary value problems for nonclassical equations of the third order $ {\mathop{\rm sgn}} x \, u_{ttt} + u_ {xx} = f (x, t), \quad {\mathop{\rm sgn}} x \, u_{t}-u_{xxx} = f (x, t)$ with changing direction time. For these problems, we prove theorems the existence and uniqueness of generalized solutions. The proof makes essential use Theorem Vishik–Lax–Milgram and the method of obtaining a priori estimates.