Geodesic
Boundary Problems for a Third-Order Equations with Changing Time Direction
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 14 (2012), pp. 19-28

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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.
Keywords: the boundary value problem, the equation of third order with a changing time direction, the generalized solutions. 
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     title = {Boundary {Problems} for a {Third-Order} {Equations} with {Changing} {Time} {Direction}},
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V. I. Antipin; S. V. Popov. Boundary Problems for a Third-Order Equations with Changing Time Direction. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 14 (2012), pp. 19-28. http://geodesic.mathdoc.fr/item/VYURU_2012_14_a1/