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Unique solvability of problems for a mixed parabolic equation in unbounded domain. (English) Zbl 1452.35113

Summary: In the work, parabolic equations with perpendicular time directions considered together and the Gevrey problem with generalized gluing conditions, and a problem with the Bitsadze-Samarskii conditions have been studied. The uniqueness and existence of the solution of the considered problems were proved. The Gevrey problem was equivalently reduced to the two-point problem for ordinary differential equation involving fractional differential operators. The uniqueness of the solution of the obtained problem was proved by the principle of extremum and the existence was proved by the method of integral equations. The theory of the second kind Volterra integral equations and properties of two-parameter Mittag-Leffler function are applied at studying a unique solvability of the problem with Bitsadze-Samarskii condition. Necessary conditions for given functions for solvability of considered problems have been found.

MSC:

35M12 Boundary value problems for PDEs of mixed type
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
45B05 Fredholm integral equations
45D05 Volterra integral equations
34A08 Fractional ordinary differential equations
34B60 Applications of boundary value problems involving ordinary differential equations
35B45 A priori estimates in context of PDEs
33E12 Mittag-Leffler functions and generalizations
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