%0 Journal Article
%T Existence and Uniqueness of the Solutions for Some Initial-Boundary Value Problems with the Fractional Dynamic Boundary Condition
%A Mykola Krasnoschok
%A Nataliya Vasylyeva
%J International Journal of Partial Differential Equations
%D 2013
%R 10.1155/2013/796430
%X 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 H£żlder 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 H£żlder classes of the solution to the corresponding nonlinear problems. 1. Introduction Let and be any numbers from and let , ; ; ; be a fixed value. In this paper, we consider a partial differential equation with the fractional derivative in time as follows: Here, denotes the Caputo fractional derivative with respect to and is defined by (see, e.g., in [1]), where is the gamma function, , , are the given functions, and is a positive. Note that if , then (1) represents a parabolic equation. As we are interested in the fractional cases, we restrict the order to the case . We will solve (1) satisfying the following conditions: the fractional dynamic boundary condition on : and one of the following conditions on : the Dirichlet boundary condition: or the Neumann boundary condition: or the fractional dynamic boundary condition: Here, and are given positive functions, and , , and , , are given functions. Concerning problems (1)¨C(6), they have the following features. First of all, (4) is a fractional dynamic boundary condition; next, these problems are formulated for the subdiffusion linear equation. Note that if , conditions (4) and (6) are called normal dynamic boundary conditions. These conditions are very natural in many mathematical models, including heat transfer in a solid in contact with a moving fluid, thermoelasticity, diffusion phenomena, and problems in fluid dynamics, and in the Stefan problem, (see [2¨C4] and the references therein). At the present moment, there are a lot of works concerning linear and nonlinear problems with dynamic boundary conditions. Here we make no pretence to provide a complete survey on the results related to problems of the type (1)-(6), if , and present only some of them. The initial-boundary value problems for the heat equation in the certain shape of domains with linear dynamic boundary condition have been solved with the separation variables method or with the Laplace transformation in [3]. In the case of smooth domains, these problems have been researched with the approaches of the general theory for evolution
%U http://www.hindawi.com/journals/ijpde/2013/796430/