No. 40 (299), issue 14Pages 19 - 28

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

V.I. Antipin, S.V. Popov
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 $ sgn x , u_{ttt} + u_ {xx} = f (x, t), quad 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.
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Keywords
the boundary value problem, the equation of third order with a changing time direction, the generalized solutions.
References
1. Kozhanov A.I. The Theory Equations of Composite Type: Thesis of Doctor of Physical and Mathematical Sciences [K teorii uravnenij sostavnogo tipa: dis. ... d-ra fiz.-mat. nauk]. Novosibirsk, 1993. 334 p.
2. Kislov N.V. Boundary Value Problems for Equations of Mixed Type in a Rectangular Region [Kraevye zadachi dlja uravnenija smeshannogo tipa v prjamougol'noj oblasti]. Dokl. AN SSSR [Reports of the Academy of Sciences of the USSR], 1980, vol. 255, no. 1, pp. 26-30.
3. Pjatkov S.G. Properties of the Functions of a Spectral Problem and Their Applications [Svojstva sobstvennyh funkcij odnoj spektral'noj zadachi i nekotorye ih prilozhenija]. Institute of Mathematics of the Siberian Branch of the USSR [AN SSSR. Sib. otd-nie. In-t matematiki], Novosibirsk, 1986, pp. 65-84.
4. Egorov I.E., Fedorov V.E. Nonclassical Equations of Mathematical Physics of High-Order [Neklassicheskie uravnenija matematicheskoj fiziki vysokogo porjadka]. Novosibirsk, VC SO RAN, 1995. 133 p.
5. Egorov I.E., Pjatkov S.G., Popov S.V. Nonclassical Differential-Operator Equations [Neklassicheskie differencial'no-operatornye uravnenija]. Novosibirsk, Nauka, 2000. 336 p.
6. Egorov Ju.V. Lectures on Partial Differential Equations. Additional Chapters [Lekcii po uravnenijam s chastnymi proizvodnymi. Dopolnitel'nye glavy]. Moscow, MGU, 1985. 166 p.
7. Cattabriga L. Potenziali di linea e di dominio per equazioni non paraboliche in due variabili a caratteristiche multiple. Rendiconti del seminario matimatico della Univ. di Padova, 1961, vol. 31, pp. 1-45.
8. Dzhuraev T.D. Boundary Value Problems for Equations of Mixed and Mixed-Composite type [Kraevye zadachi dlja uravnenija smeshannogo i smeshanno-sostavnogo tipov]. Tashkent, FAN, 1979. 239 p.
9. Antipin V.I. Solvability of Boundary Value Problems for the Third Order with Changing Time Direction [Razreshimost' kraevoj zadachi dlja uravnenija tret'ego porjadka s menjajuwimsja napravleniem vremeni]. Matematicheskie zametki JaGU [Mathematical Notes YSU], 2011, vol. 18, no. 1, pp. 8-15.
10. Cattabriga L. Un Problema al cjntorno per una equazione parabolica di ordine dispari. Annali della Scuola Normale Superiore di Pisa, 1959, vol. 13, no. 2, pp. 163-203.