Table of Contents
International Journal of Partial Differential Equations
Volume 2013 (2013), Article ID 796430, 20 pages
http://dx.doi.org/10.1155/2013/796430
Research Article

Existence and Uniqueness of the Solutions for Some Initial-Boundary Value Problems with the Fractional Dynamic Boundary Condition

Institute of Applied Mathematics and Mechanics of NAS of Ukraine, R.Luksemburg Street 74, Donetsk 83114, Ukraine

Received 22 April 2013; Accepted 28 July 2013

Academic Editor: Antonin Novotny

Copyright © 2013 Mykola Krasnoschok and Nataliya Vasylyeva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we analyze some initial-boundary value problems for the subdiffusion equation with a fractional dynamic boundary condition in a one-dimensional bounded domain. First, we establish the unique solvability in the Hölder space of the initial-boundary value problems for the equation , , where L is a uniformly elliptic operator with smooth coefficients with the fractional dynamic boundary condition. Second, we apply the contraction theorem to prove the existence and uniqueness locally in time in the Hölder classes of the solution to the corresponding nonlinear problems.