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

Abstract EN

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 ∂tαu(x,t)=Lu(x,t)+f0(x,t), α∈(0,1), 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.