On the solvability of an initial-boundary value problem for a non-linear fractional diffusion equation

Abstract

<abstract><p>In this paper, we consider an initial-boundary value problem for a non-linear fractional diffusion equation on a bounded domain. The fractional derivative is defined in Caputo's sense with respect to the time variable and represents the case of sub-diffusion. Also, the equation involves a second order symmetric uniformly elliptic operator with time-independent coefficients. These initial-boundary value problems arise in applied sciences such as mathematical physics, fluid mechanics, mathematical biology and engineering. By using eigenfunction expansions and Banach fixed point theorem, we establish the existence, uniqueness and regularity properties of the solution of the problem.</p></abstract>

Investigation of Well-Posedness for a Direct Problem for a Nonlinear Fractional Diffusion Equation and an Inverse Problem

“…In this study, we first consider a direct problem for a nonlinear time-fractional partial differential equation with initial and boundary conditions. We study the well-posedness of the problem by the methodology of [15]. Then, we apply the results to an inverse problem with additional data and we obtain the stability of the solution (u, r).…”

“…by using the hypothesis (52), inequality (15), Lemma 1 with the properties of H s (Ω) spaces. From (57) and the properties of {φ n } ∞ n=1 , we write…”

“…It was introduced by [7,8] for a homogeneous equation, further investigated by [9] and was applied to an inhomogeneous case by [6]. Then, the technique for examining the existence, uniqueness and regularity of the solution for a nonlinear model was developed by [10] and generalized by [11][12][13][14][15]. We note that the last five works employ a contraction mapping method by assuming the solution as the fixed point.…”